22:
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751:
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essentially unobservable. They can be detected only by the discontinuities that may occur in demand with variation in price-ratios, leading to an abrupt jumping of a point of tangency across a chasm when the straight line is rotated. But, while such discontinuities may reveal the existence of chasms, they can never measure their depth. The concave portions of the indifference curves and their many-dimensional generalizations, if they exist, must forever remain in unmeasurable obscurity.
11766:
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16519:
11316:
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11523:"in the aggregate, the discrepancy between an allocation in the fictitious economy generated by and some allocation in the real economy is bounded in a way that is independent of the number of economic agents. Therefore, the average agent experiences a deviation from intended actions that vanishes in significance as the number of agents goes to infinity".
11063:
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It will be noted that any point where the indifference curves are convex rather than concave cannot be observed in a competitive market. Such points are shrouded in eternal darkness—unless we make our consumer a monopsonist and let him choose between goods lying on a very convex "budget curve" (along
11337:
optimal-baskets. For example, we can imagine that, for zoos, a lion costs as much as an eagle, and further that a zoo's budget suffices for one eagle or one lion. We can suppose also that a zoo-keeper views either animal as equally valuable. In this case, the zoo would purchase either one lion or one
11531:
summarized their economic implications: "Some key results obtained under the convexity assumption remain (approximately) relevant in circumstances where convexity fails. For example, in economies with a large consumption side, preference nonconvexities do not destroy the standard results".
446:
and its convex hull is only 1/4, which is half the distance (1/2) between its summand {0, 1} and . As more sets are added together, the average of their sum "fills out" its convex hull: The maximum distance between the average and its convex hull approaches zero as the
12223:
convex hull; this equality implies that the
Shapley–Folkman–Starr results are useful in probability theory. In the other direction, probability theory provides tools to examine convex sets generally and the Shapley–Folkman–Starr results specifically. The Shapley–Folkman–Starr results have been
12168:
problems, despite the non-convexities of the summand functions. Ekeland and later authors argued that additive separability produced an approximately convex aggregate problem, even though the summand functions were non-convex. The crucial step in these publications is the use of the
Shapley–Folkman
11453:, who was then an undergraduate enrolled in Arrow's (graduate) advanced mathematical-economics course. In his term-paper, Starr studied the general equilibria of an artificial economy in which non-convex preferences were replaced by their convex hulls. In the convexified economy, at each price, the
11106:
Following Starr's 1969 paper, the
Shapley–Folkman–Starr results have been widely used to show that central results of (convex) economic theory are good approximations to large economies with non-convexities; for example, quasi-equilibria closely approximate equilibria of a convexified economy.
11272:
is defined for each consumer; a consumer's indifference curve contains all the baskets of commodities that the consumer regards as equivalent: That is, for every pair of baskets on the same indifference curve, the consumer does not prefer one basket over another. Through each basket of commodities
13931:
The concept of a convex set (i.e., a set containing the segment connecting any two of its points) had repeatedly been placed at the center of economic theory before 1964. It appeared in a new light with the introduction of integration theory in the study of economic competition: If one
11357:
If indifference curves for purchases be thought of as possessing a wavy character, convex to the origin in some regions and concave in others, we are forced to the conclusion that it is only the portions convex to the origin that can be regarded as possessing any importance, since the others are
11102:
economy, in which non-convex sets were replaced by their convex hulls; Starr proved that the convexified economy has equilibria that are closely approximated by "quasi-equilibria" of the original economy; moreover, he proved that every quasi-equilibrium has many of the optimal properties of true
12160:. Lemaréchal's problem was additively separable, and each summand function was non-convex; nonetheless, a solution to the dual problem provided a close approximation to the primal problem's optimal value. Ekeland's analysis explained the success of methods of convex minimization on
10947:
11296:
a consumer's preference set, as shown in the diagram. This means that an optimal basket is on the highest possible indifference curve given the budget-line, which is defined in terms of a price vector and the consumer's income (endowment vector). Thus, the set of optimal baskets is a
3704:
9277:
5564:
The circumradius (blue) and inner radius (green) of a point set (dark red, with its convex hull shown as the lighter red dashed lines). The inner radius is smaller than the circumradius except for subsets of a single circle, for which they are
8016:
6079:
41:) is the sum of four points (+) from the (left-hand) sets—two points in two non-convex sets plus two points in the convex hulls of two sets. The convex hulls are shaded pink. The original sets each have exactly two points (shown as red dots).
1874:
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The squared
Euclidean distance is a measure of how "close" two sets are. In particular, if two sets are compact, then their squared Euclidean distance is zero if and only if they are equal. Thus, we may quantify how close to convexity
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which he is affecting the price of what he buys). In this monopsony case, we could still deduce the slope of the man's indifference curve from the slope of the observed constraint at the equilibrium point.
10006:
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409:
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13236:, p. 26): "After all, one may be indifferent between an automobile and a boat, but in most cases one can neither drive nor sail the combination of half boat, half car."
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709:. For finite-dimensional vector spaces, the operations of vector addition and real-number multiplication can each be defined coordinate-wise, following the example of the Cartesian plane.
9485:
8784:
4244:
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14830:. Classics in Applied Mathematics. Vol. 28 (Corrected reprinting of the North-Holland ed.). Philadelphia: Society for Industrial and Applied Mathematics (SIAM). pp. 357–373.
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3919:{\displaystyle \mathrm {Conv} \left(\sum _{n=1}^{N}Q_{n}\right)\subseteq \bigcup _{I\subseteq \{1,2,\ldots N\}:~|I|=D}\left(\sum _{n\in I}\mathrm {Conv} (Q_{n})+\sum _{n\notin I}Q_{n}\right).}
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1503:
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8353:
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between the set of equilibria of the "convexified" economy and the set of quasi-equilibria of the original economy followed from Starr's corollary to the
Shapley–Folkman theorem.
10542:
2340:
11476:" of the original economy, when the number of agents exceeds the dimension of the goods: Concretely, Starr proved that there exists at least one quasi-equilibrium of prices
11472:
In his 1969 publication, Starr applied the
Shapley–Folkman–Starr theorem. Starr proved that the "convexified" economy has general equilibria that can be closely approximated by "
11266:
are defined over all "baskets" of goods. Each basket is represented as a non-negative vector, whose coordinates represent the quantities of the goods. On this set of baskets, an
7407:
5958:
15259:
14203:
Aubin, Jean-Pierre (2007). "14.2 Duality in the case of non-convex integral criterion and constraints (especially 14.2.3 The
Shapley–Folkman theorem, pages 463–465)".
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8858:
6841:
6699:
6388:
5644:
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11166:
The
Shapley–Folkman lemma enables researchers to extend results for Minkowski sums of convex sets to sums of general sets, which need not be convex. Such sums of sets arise in
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9084:
8953:
7692:
6255:
This can be interpreted as stating that, as long as we have an upper bound on the inner radii, performing "Minkowski-averaging" would get us closer and closer to a convex set.
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in 1974 to explain the apparent convexity of separable problems with many summands, despite the non-convexity of the summand problems. In 1973, the young mathematician
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Of course, the given optimal-point is a sum of points in the graphs of the original summands and of a small number of convexified summands, by the
Shapley–Folkman lemma.
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7179:
1302:
1250:
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has been extended by
Artstein. Different proofs have also appeared in unpublished papers. An elementary proof of the Shapley–Folkman lemma can be found in the book by
883:
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245:
198:
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224:
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15013:
Contributions to Operations Research and Economics: The twentieth anniversary of CORE (Papers from the symposium held in Louvain-la-Neuve, January 1987)
10161:
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1639:
This operation is clearly commutative and associative on the collection of non-empty sets. All such operations extend in a well-defined manner to recursive forms
421: , which is convex. The Shapley–Folkman lemma implies that every point in is the sum of an integer from {0, 1} and a real number from .
11307:. If the preference set is convex, then at every price the consumer's demand is a convex set, for example, a unique optimal basket or a line-segment of baskets.
2053:
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8713:
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7649:
7587:
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6448:
6324:
6248:{\displaystyle d^{2}\left(\mathrm {Conv} \left({\frac {1}{N}}\sum _{n=1}^{N}Q_{n}\right),{\frac {1}{N}}\sum _{n=1}^{N}Q_{n}\right)\leq {\frac {Dr_{0}^{2}}{N}}.}
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1934:
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1096:
of two convex sets is a convex set. More generally, the intersection of a family of convex sets is a convex set. In particular, the intersection of two
10166:
12549:, pp. 49 and 75). Ensuring that the Minkowski sum of sets be closed requires the closure operation, which appends limits of convergent sequences.
7274:
7059:
15080:. Grundlehren der Mathematischen Wissenschaften . Vol. 306. Berlin: Springer-Verlag. pp. 136–193 (and bibliographical comments on pp. 334–335).
5047:
17208:
11342:)! Thus, the zoo-keeper's preferences are non-convex: The zoo-keeper prefers having either animal to having any strictly convex combination of both.
5148:
4298:
283:
143:
to non-convex preferences. In optimization theory, it can be used to explain the successful solution of minimization problems that are sums of many
14678:
14570:
11532:"The derivation of these results in general form has been one of the major achievements of postwar economic theory", wrote Guesnerie. The topic of
1946:
10942:{\displaystyle Var=Var\left=\sum _{n\in I}Var\left\leq \sum _{n\in I}\left(Var(Q_{n})+\epsilon \right)\leq \sum _{\max D}Var(Q_{n})+D\epsilon .}
9282:
5791:
16239:
8166:
12169:
lemma. The Shapley–Folkman lemma has encouraged the use of methods of convex minimization on other applications with sums of many functions.
2625:
12692:, pp. 357–359): Published in the first English edition of 1976, Ekeland's appendix proves the Shapley–Folkman lemma, also acknowledging
4163:
3157:
885:; the non-convex set of three integers {0, 1, 2} is contained in the interval , which is convex. For example, a solid
11656:). The Shapley–Folkman lemma has encouraged the use of methods of convex minimization on other applications with sums of many functions.
11091:
12446:
16238:. (Draft of second edition, from Starr's course at the Economics Department of the University of California, San Diego). Archived from
14038:. Advanced Textbooks in Economics. Vol. 12 (reprint of San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts
17412:
12205:
6391:
17056:
1642:
9933:
3040:
16555:
15851:
15353:
14720:(Presidential address delivered at the 103rd meeting of the American Economic Association, 29 December 1990, Washington, DC): 1–7.
3411:
1511:
15809:(1969). "Quasi-equilibria in markets with non-convex preferences (Appendix 2: The Shapley–Folkman theorem, pp. 35–37)".
14773:
17312:
14116:
12271:, which bounds the volume of sums in terms of the volumes of their summand-sets. The volume of a set is defined in terms of the
10603:
10062:
8296:
7184:
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17404:
16016:
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15762:
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15050:
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14797:
14212:
13098:
4408:
11281:
of the indifference curve and all the commodity baskets that the consumer prefers over the indifference curve. A consumer's
9407:
16504:
15011:(1989). "First-best allocation of resources with nonconvexities in production". In Cornet, Bernard; Tulkens, Henry (eds.).
11387:, and Rothenberg. In particular, Rothenberg's paper discussed the approximate convexity of sums of non-convex sets. These
11107:"The derivation of these results in general form has been one of the major achievements of postwar economic theory", wrote
12479:
11878:
17678:
16145:; Juréen, Lars (in association with Wold) (1953). "8 Some further applications of preference fields (pp. 129–148)".
15250:(April 1973). Utilisation de la dualité dans les problémes non convexes (Report) (in French). Domaine de Voluceau,
6513:
5572:
9750:
6707:
17417:
16306:
14735:
14232:
8718:
12156:
need not provide useful information for solving the primal problem, unless the primal problem be convex and satisfy a
10980:
9272:{\displaystyle Var(S):=\sup _{x\in \mathrm {Conv} (S)}\inf _{\mathbb {E} =x,X{\text{ is finitely supported in }}S}Var}
1434:
16225:
15982:
15892:
15701:
15635:
15499:
15371:
15315:
15205:
15171:
15085:
15020:
14990:
14835:
14654:
14580:
14479:
14457:
14128:
14047:
13603:
12729:
10013:
8863:
6453:
4249:
3508:
2765:
11597:
11399:, which considered convexified consumer-preferences and introduced the concept of an "approximate equilibrium". The
16065:"An application of the central limit theorem for Banach-space–valued random variables to the theory of random sets"
14969:
Green, Jerry; Heller, Walter P. (1981). "1 Mathematical analysis and convexity with applications to economics". In
14882:
14065:
Artstein, Zvi (1980). "Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points".
12259:, and the Shapley–Folkman lemma has applications in non-probabilistic measure theory, such as the theories of
11375:
5508:
4573:
6283:, together with applications in estimating the duality gap in separable optimization problems and zero-sum games.
4900:
1161:. The convex hull of a set can be equivalently defined to be the set of all convex combinations of points in
17653:
17437:
17354:
14940:
Farrell, M. J. (October 1961b). "The Convexity assumption in the theory of competitive markets: Rejoinder".
14207:(Reprint with new preface of 1982 North-Holland revised English ed.). Mineola, NY: Dover Publications.
12737:
12245:
4083:
13117:(1981). "Approximation of points of convex hull of a sum of sets by points of the sum: An elementary approach".
11338:
eagle. Of course, a contemporary zoo-keeper does not want to purchase half of an eagle and half of a lion (or a
4967:
11527:
Following Starr's 1969 paper, the Shapley–Folkman–Starr results have been widely used in economic theory.
5747:
5367:
424:
The distance between the convex interval and the non-convex set {0, 1, 2} equals one-half
13952:
over a collection of insignificant agents is an insight that economic theory owes ... to integration theory.
12318:
4645:
17693:
15696:. Encyclopedia of Mathematics and its Applications. Vol. 44. Cambridge, UK: Cambridge University Press.
11319:
When the consumer's preferences have concavities, the consumer may jump between two separate optimal baskets.
9548:
5422:
4013:
Shapley and Folkman used their lemma to prove the following theorem, which quantifies the difference between
2893:
2719:
15450:
Puri, Madan L.; Ralescu, Dan A. (1985). "Limit theorems for random compact sets in Banach space".
14649:. London Mathematical Society lecture note series. Vol. 62. Cambridge, UK: Cambridge University Press.
14533:
Proceedings of 1981 IEEE Conference on Decision and Control, San Diego, CA, December 1981, pp. 432–443.
14470:(1996). "5.6 Large scale separable integer programming problems and the exponential method of multipliers".
11373:
Nonetheless, non-convex preferences were illuminated from 1959 to 1961 by a sequence of papers in
7697:
4500:
4118:
2966:
2825:
2501:
702:
500:
17668:
16747:
16548:
12197:
11866:
17457:
15388:; Whinston, Michael D.; Green, Jerry R. (1995). "17.1 Large economies and nonconvexities".
10471:
2283:
17041:
16993:
16681:
15341:
12596:
12268:
11370:, who wrote that non-convexities are "shrouded in eternal darkness ...", according to Diewert.
5899:
5320:
Note that this upper bound depends on the dimension of ambient space and the shapes of the summands, but
483:
16171:
14602:
13057:
12248:
used the Shapley–Folkman–Starr results to avoid the assumption that all the random sets be convex.
12069:", where convex hulls are taken of the graphs of the summand functions. Such an optimal solution is the
10420:
8823:
8011:{\displaystyle ({\bar {x}})_{D+n_{0}}=\left(\sum _{n}\sum _{k}w'_{n,k}{\bar {q}}_{n,k}\right)_{D+n_{0}}}
6811:
6655:
6358:
4036:
3368:
2577:
17422:
16703:
16691:
16686:
16676:
15045:. Princeton studies in mathematical economics. Vol. 5. Princeton, NJ: Princeton University Press.
12358:
11178:; in each of these three mathematical sciences, non-convexity is an important feature of applications.
9492:
9031:
8912:
2178:
84:
12567:
12280:
11494:, all consumers can choose optimal baskets (maximally preferred and meeting their budget constraints).
11469:
lemma and theorem in "private correspondence", which was reported by Starr's published paper of 1969.
11345:
When the consumer's preference set is non-convex, then (for some prices) the consumer's demand is not
9689:
7005:
6782:
6397:
6329:
6306:
The following proof of Shapley–Folkman lemma is from. The proof idea is to lift the representation of
5963:
2450:
2367:
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1307:
17462:
17121:
16646:
16597:
12600:
12576:
11533:
11442:
11324:
11171:
11115:
9875:
5697:
3640:
2398:
1092:
132:
14880:
Farrell, M. J. (August 1959). "The Convexity assumption in the theory of competitive markets".
12502:, p. 169), "Markets with non-convex preferences and production", which presents the results of
10954:
10547:
9089:
7370:
5270:
3600:
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1887:
1402:
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17427:
17269:
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16656:
16338:
15416:
14108:
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11537:
11457:
was the sum of convex hulls of the consumers' demands. Starr's ideas interested the mathematicians
11423:
11119:
6616:
5618:
3568:
2239:
1884:
Minkowski addition behaves well with respect to taking convex hulls. Specifically, for all subsets
479:
15839:
15337:
11600:(for economists). The Shapley–Folkman–Starr results have also influenced economics research using
7654:
7595:
17512:
17489:
17383:
17307:
17189:
17086:
17081:
17031:
16904:
16782:
16541:
16499:
15919:
Markets, information and uncertainty: Essays in economic theory in honor of Kenneth J. Arrow
14230:
Aubin, J. P.; Ekeland, I. (1976). "Estimates of the duality gap in nonconvex optimization".
13589:
12237:
8133:
5992:
5041:
698:
83:
The lemma may be intuitively understood as saying that, if the number of summed sets exceeds the
15270:(1978). "A note on the core equivalence theorem: How many blocking coalitions are there?".
11504:
in the convexified economy, every good's market is in equilibrium: Its supply equals its demand.
8958:
7840:
7791:
7155:
1869:{\displaystyle \sum _{n=1}^{N}Q_{n}=\{\sum _{n=1}^{N}q_{n}\mid q_{n}\in Q_{n},~1\leq n\leq N\}.}
1255:
1203:
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15977:. Lecture Notes in Economics and Mathematical Systems. Vol. 223. Berlin: Springer-Verlag.
15487:
15251:
14350:(January 1966). "Existence of competitive equilibrium in markets with a continuum of traders".
12733:
12586:
12572:
12426:
11745:
11668:
11621:
11593:
11561:
11507:
For each quasi-equilibrium, the prices "nearly clear" the markets for the original economy: an
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11191:
11143:
7752:
1170:
971:
414:
144:
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14784:. Handbooks in Economics. Vol. 2. Amsterdam: North-Holland Publishing. pp. 535–599.
12589:
of the original problem. Their study of duality gaps was extended by Di Guglielmo to the
11977:
are separable. Given a separable problem with an optimal solution, we fix an optimal solution
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8443:
8381:
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230:
183:
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15914:
15879:
Starr, Ross M. (1997). "8 Convex sets, separation theorems, and non-convex sets in
15218:(October 1961). "Convexity assumptions, allocative efficiency, and competitive equilibrium".
12495:
12430:
12256:
12233:
12149:
11664:
11641:
11601:
11576:(1987). The Shapley–Folkman–Starr results have been featured in the economics literature: in
11403:-papers and the Shapley–Shubik paper influenced another notion of "quasi-equilibria", due to
11349:; a disconnected demand implies some discontinuous behavior by the consumer, as discussed by
11293:
11231:
4005:
In particular, the Shapley–Folkman lemma requires the vector space to be finite-dimensional.
3301:
960:
706:
570:
273:
15408:
14977:. Handbooks in Economics. Vol. 1. Amsterdam: North-Holland Publishing. pp. 15–52.
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associates with every agent of an economy an arbitrary set in the commodity space and
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10339:
8789:
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2165:{\displaystyle \mathrm {Conv} (\sum _{n=1}^{N}Q_{n})=\sum _{n=1}^{N}\mathrm {Conv} (Q_{n})}
148:
16181:. Berkeley, Calif.: Economics Department, University of California, Berkeley. pp. 1–5
15794:
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15069:
14187:
13067:. Berkeley, Calif.: Economics Department, University of California, Berkeley. pp. 1–5
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8:
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16792:
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16440:
16435:
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16326:
16034:
Vind, Karl (May 1964). "Edgeworth-allocations in an exchange economy with many traders".
15595:
15409:
15377:
14134:
13996:
12741:
12516:
12314:
12284:
12252:
12225:
12142:
12070:
11676:
11653:
11629:
10140:
9634:
8417:
8355:
It follows that there is at least one element of the sum on the r.h.s. that is non-zero.
7412:
1154:
152:
48:
15528:
Rothenberg, Jerome (October 1960). "Non-convexity, aggregation, and Pareto optimality".
15463:
15166:. Economic theory, econometrics, and mathematical economics. New York: Academic Press .
14117:"22 Discrete and continuous bang–bang and facial spaces or: Look for the extreme points"
5669:
17531:
17249:
17026:
16972:
16899:
16809:
16767:
16757:
16725:
16720:
16666:
16661:
16466:
16425:
16348:
16051:
16005:
15826:
15782:
15691:
15670:
15574:
15545:
15235:
15123:
15038:
14957:
14928:
14911:
Farrell, M. J. (October 1961a). "On Convexity, efficiency, and markets: A Reply".
14899:
14752:
14721:
14613:(1975). "Measures of the non-convexity of sets and the Shapley–Folkman–Starr theorem".
14433:
14416:
Bator, Francis M. (October 1961b). "On convexity, efficiency, and markets: Rejoinder".
14404:
14367:
14289:
14249:
14167:
14082:
13940:. But explanations of the ... functions of prices ... can be made to rest on the
12520:
12438:
12178:
12001:
11974:
11925:
11830:
11569:
11512:
11434:
11278:
11268:
11195:
11175:
11151:
9730:
9669:
9614:
9594:
9129:
9011:
8991:
8698:
8678:
8479:
8361:
7820:
7634:
7572:
7034:
6596:
6433:
6309:
5649:
5488:
5468:
5402:
5300:
4880:
4074:
4016:
3984:
3964:
3944:
3678:
3020:
2557:
2479:
2347:
1919:
1394:
1376:
1197:
1193:
1117:
979:
655:
633:
608:
140:
136:
106:
57:
26:
14982:
14789:
10300:{\displaystyle \{1,2,...,N\},|I|\leq D,x_{n}\in \mathrm {Conv} (Q_{n}),q_{n}\in Q_{n}}
3333:
17349:
17159:
17147:
17126:
17096:
16958:
16882:
16838:
16522:
16410:
16368:
16358:
16292:
16221:
16012:
15978:
15930:
15888:
15865:
15847:
15843:
15732:
15697:
15631:
15495:
15428:
15393:
15385:
15367:
15333:
15311:
15299:
15283:
15267:
15215:
15201:
15193:
15189:
15167:
15146:
15081:
15046:
15016:
14986:
14866:
14831:
14793:
14650:
14576:
14549:
14498:
14475:
14453:
14330:
14313:
14208:
14124:
14043:
13761:
13599:
13173:
13130:
13094:
12610:
11649:
11549:
11415:
11384:
11131:
6287:
6280:
626:
61:
17688:
17502:
17447:
17338:
17333:
17111:
17091:
16998:
16948:
16919:
16777:
16742:
16639:
16617:
16373:
16213:
16126:
16122:
16080:
16043:
15953:
15922:
15857:
15818:
15790:
15774:
15724:
15662:
15604:
15566:
15537:
15467:
15420:
15359:
15279:
15227:
15135:
15115:
15103:
14978:
14949:
14920:
14891:
14858:
14785:
14744:
14687:
14642:
14622:
14610:
14598:
14537:
14517:
14425:
14396:
14359:
14325:
14281:
14241:
14183:
14157:
14074:
13165:
13126:
12606:
12582:
12322:
12272:
12145:
11581:
11454:
11350:
6295:
6272:
15798:
15139:
14733:
Di Guglielmo, F. (1977). "Nonconvex duality in multiobjective optimization".
14709:
14692:
14673:
13921:
11545:
11127:
8668:{\displaystyle {\bar {x}}=\sum _{n}\left(\sum _{k}w'_{n,k}{\bar {q}}_{n,k}\right)}
459:
The Shapley–Folkman lemma depends upon the following definitions and results from
17522:
17493:
17467:
17388:
17373:
17282:
17254:
17231:
17179:
16651:
16588:
16492:
16430:
16363:
16262:
16231:
16150:
16130:
16090:
16022:
15988:
15961:
15898:
15742:
15707:
15678:
15614:
15515:
15505:
15494:. Princeton Landmarks in Mathematics. Princeton, NJ: Princeton University Press.
15475:
15438:
15321:
15287:
15177:
15091:
15056:
15026:
15008:
14996:
14841:
14803:
14760:
14697:
14660:
14630:
14586:
14557:
14485:
14375:
14335:
14297:
14257:
14218:
14175:
14090:
14053:
13769:
13134:
12276:
12194:
11821:
11774:
11645:
11625:
11528:
11346:
11108:
7269:
By Carathéodory's theorem for conic hulls, we have an alternative representation
1142:
886:
460:
434:
53:
15145:(Report). Cowles Foundation discussion papers. Vol. 538. New Haven, Conn.:
14502:
490:
of real numbers, called "coordinates", which are conventionally denoted by
21:
17609:
17517:
17378:
17277:
17154:
16894:
16853:
16772:
16564:
16390:
15913:; Stinchcombe, M. B. (1999). "Exchange in a network of trading posts". In
15861:
15650:
15363:
15345:
14501:; Lauer, Gregory S.; Sandell, Nils R. Jr.; Posbergh, Thomas A. (January 1983).
14191:
13441:
Taking the convex hull of non-convex preferences had been discussed earlier by
12442:
12434:
12288:
12279:. In advanced measure-theory, the Shapley–Folkman lemma has been used to prove
12264:
12190:
11846:
11585:
11577:
11565:
11557:
11367:
11259:
11147:
11139:
5257:{\displaystyle d^{2}(Q,\mathrm {Conv} (Q))~\leq ~\sum _{\max D}rad(Q_{n})^{2}.}
750:
722:
263:
15728:
15609:
15590:
15471:
14626:
8817:
The following "probabilistic" proof of Shapley–Folkman–Starr theorem is from.
6263:
There have been many proofs of these results, from the original, to the later
17647:
17393:
17076:
16976:
16967:
16938:
16924:
16914:
16858:
16622:
16612:
16602:
16217:
15926:
15758:
15754:
15586:
15415:. Probability and its applications. London: Springer-Verlag London. pp.
14970:
14862:
14777:
14553:
14521:
14347:
14309:
14269:
14162:
14145:
14121:
Game and economic theory: Selected contributions in honor of Robert J. Aumann
14104:
14027:
13765:
13723:, p. 38) Lemaréchal's experiments were discussed in later publications:
13177:
13114:
12475:
12470:
12295:" (alternatively, "image") is the set of values produced by the function. A
12241:
11541:
11458:
11446:
11427:
11419:
11404:
11396:
11392:
11123:
11095:
11079:
11067:
7562:{\displaystyle {\bar {x}}=(x,1,...,1);\quad {\bar {q}}_{n,k}=(q_{n,k},e_{n})}
6968:{\displaystyle {\bar {x}}=(x,1,...,1);\quad {\bar {q}}_{n,k}=(q_{n,k},e_{n})}
6264:
3941:
If a vector space obeys the Shapley–Folkman lemma for a natural number
1097:
931:
898:
69:
14387:
Bator, Francis M. (October 1961a). "On convexity, efficiency, and markets".
12575:
of a problem of non-convex minimization—that is, the problem defined as the
12538:
Clos(P) + Clos(Q) ⊆ Clos( Clos(P) + Clos(Q) );
1940:
of their Minkowski sum is the Minkowski sum of their convex hulls. That is,
428:
1/2 = |1 − 1/2| = |0 − 1/2| = |2 − 3/2| = |1 − 3/2|.
87:
of the vector space, then their Minkowski sum is approximately convex.
17614:
17046:
16878:
16873:
16607:
16456:
15310:. Econometric Society monographs. Vol. 9. Cambridge University Press.
14815:
13992:
12725:
12153:
12134:
11573:
11553:
11155:
11135:
6286:
Usual proofs of these results are nonconstructive: they establish only the
6276:
2037:{\displaystyle \mathrm {Conv} (A+B)=\mathrm {Conv} (A)+\mathrm {Conv} (B).}
865:
is not, because it does not contain a line segment joining its points
806:
778:
765:
737:
487:
475:
448:
168:
65:
16203:(Section 8.2.3 Measuring non-convexity, the Shapley–Folkman theorem)"
15424:
14452:(Second ed.). Cambridge, Mass.: Athena Scientific. pp. 494–498.
9391:{\displaystyle d(S,\mathrm {Conv} (S))^{2}\leq Var(S)\leq r(S)\leq rad(S)}
5885:{\displaystyle d^{2}(Q,\mathrm {Conv} (Q))\leq \sum _{\max D}r(Q_{n})^{2}}
17604:
17599:
17483:
17071:
16983:
16863:
16715:
16629:
16142:
16102:
15944:
Tardella, Fabio (1990). "A new proof of the Lyapunov convexity theorem".
14823:
14748:
14245:
14100:
12603:
12579:
12532:
12465:
12445:
counterpart of the Shapley–Folkman lemma, which has itself been called a
11589:
11508:
11462:
11363:
11083:
11071:
8286:{\displaystyle w'_{n_{0},k}({\bar {q}}_{n_{0},k})_{D+n_{0}}=w'_{n_{0},k}}
1937:
1178:
1174:
1131:
1113:
840:
472:
418:
202:
110:
102:
73:
30:
15975:
Market demand: An analysis of large economies with nonconvex preferences
13876:, pp. 243–244) uses applications of the Shapley–Folkman lemma from
13754:
Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences
13153:
11062:
8123:{\displaystyle =\sum _{n}\sum _{k}w'_{n,k}({\bar {q}}_{n,k})_{D+n_{0}}.}
2709:{\displaystyle \mathrm {Conv} (Q)=\sum _{n=1}^{N}\mathrm {Conv} (Q_{n})}
413:
The subset of the integers {0, 1, 2} is contained in the
17527:
17497:
17259:
16831:
16583:
16258:
16194:
16085:
16064:
16055:
16000:
15910:
15830:
15806:
15786:
15674:
15578:
15549:
15255:
15239:
15127:
14961:
14932:
14903:
14756:
14725:
14594:
14437:
14408:
14371:
14293:
14253:
14171:
14086:
14031:
13169:
12528:
12524:
12299:
is a vector-valued generalization of a measure; for example, if
11633:
11450:
11431:
11362:
The difficulties of studying non-convex preferences were emphasized by
11087:
6268:
801:
782:
757:
729:
280:
zero and one to itself yields the set consisting of zero, one, and two:
77:
34:
1177: , which contains the integer end-points. The convex hull of the
889:
is convex; however, anything that is hollow or dented, for example, a
17142:
16909:
16826:
16821:
16634:
16461:
16179:
Economics 201B: Nonconvex preferences and approximate equilibria
13065:
Economics 201B: Nonconvex preferences and approximate equilibria
12613:
12182:
11167:
7360:{\displaystyle {\bar {x}}=\sum _{n}\sum _{k}w'_{n,k}{\bar {q}}_{n,k}}
7145:{\displaystyle {\bar {x}}=\sum _{n}\sum _{k}w_{n,k}{\bar {q}}_{n,k}.}
6291:
1182:
894:
814:
792:
176:
128:
120:
16047:
15957:
15822:
15778:
15666:
15119:
14363:
14285:
14123:. Ann Arbor, Mich.: University of Michigan Press. pp. 449–462.
14078:
5129:{\displaystyle rad(S)\equiv \inf _{x\in R^{N}}\sup _{y\in S}\|x-y\|}
3291:{\displaystyle =+=\mathrm {Conv} (\{0,1\})+\mathrm {Conv} (\{0,1\})}
17536:
16934:
16929:
16804:
16383:
16315:
16149:. Wiley publications in statistics. New York: John Wiley and Sons.
15856:(Second ed.). Palgrave Macmillan. pp. 317–318 (1st ed.).
15653:(November 1950). "The problem of integrability in utility theory".
15570:
15541:
15231:
14953:
14924:
14895:
14429:
14400:
11605:
6294:
for computing the representation. In 1981, Starr published an
890:
442:
1/2 ( {0, 1} + {0, 1} ) = {0, 1/2, 1}
12040:
For this separable problem, we also consider an optimal solution
11616:
605:
More generally, any real vector space of (finite) dimension
227:
is not, because the line segment joining two distinct points
17317:
16988:
16887:
16799:
16533:
16378:
16199:"8 Convex sets, separation theorems, and non-convex sets in
14540:; O'Brien, R. C. (1978). "Cancellation characterizes convexity".
11339:
11241:, is optimal and also feasible, unlike any basket lying on
4158:
1166:
1108:
277:
97:
16069:
Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
15557:
Rothenberg, Jerome (October 1961). "Comments on non-convexity".
11410:
17174:
16963:
15921:. Cambridge, UK: Cambridge University Press. pp. 217–234.
15452:
Mathematical Proceedings of the Cambridge Philosophical Society
14615:
Mathematical Proceedings of the Cambridge Philosophical Society
12260:
11466:
11303:
6298:
for a less sharp version of the Shapley–Folkman–Starr theorem.
5560:
3365:
Shuffling indices if necessary, this means that every point in
1728:{\displaystyle \sum _{n=1}^{N}Q_{n}=Q_{1}+Q_{2}+\ldots +Q_{N}.}
172:
15763:"Quasi-cores in a monetary economy with nonconvex preferences"
10001:{\displaystyle Var\left(Q\right)\leq \sum _{\max D}Var(Q_{n})}
4396:{\displaystyle d^{2}(S,S')=\inf _{x\in S,y\in S'}\|x-y\|^{2}.}
3103:{\displaystyle q_{n}\in \mathrm {Conv} (Q_{n})\setminus Q_{n}}
404:{\displaystyle \{0,1\}+\{0,1\}=\{0+0,0+1,1+0,1+1\}=\{0,1,2\}.}
16848:
16730:
16710:
16487:
15883:(new chapters 22 and 25–26 in (2011) second ed.)".
14497:
14272:(January–April 1964). "Markets with a continuum of traders".
12747:
11640:
The Shapley–Folkman lemma has been used to explain why large
11333:, then some prices determine a budget-line that supports two
8476:
that are nonzero, we conclude that there can only be at most
7569:
this alternative representation is also a representation for
3495:{\displaystyle x=\sum _{n=1}^{D}q_{n}+\sum _{n=D+1}^{N}q_{n}}
1629:{\displaystyle \{0,1\}+\{0,1\}=\{0+0,0+1,1+0,1+1\}=\{0,1,2\}}
1057:
for some indexed set of non-negative real numbers {
630:
17230:
16268:. pp. 1–3. (Draft of article for the second edition of
14503:"Optimal short-term scheduling of large-scale power systems"
13916:, pp. 168 and 175) was noted by the winner of the 1983
12212:, the collection of the expected values of the simple,
11315:
11103:
equilibria, which are proved to exist for convex economies.
1373:
In any vector space (or algebraic structure with addition),
139:. In economics, it can be used to extend results proved for
17005:
16284:
15887:(1st ed.). Cambridge, UK: Cambridge University Press.
15192:(1957). "Allocation of resources and the price system". In
11862:
11770:
15262:, Laboratoire de recherche en informatique et automatique.
15140:
On the tendency toward convexity of the vector sum of sets
11301:
of the prices, and this function is called the consumer's
10059:, by Shapley–Folkman lemma, there exists a representation
15106:(January 1935). "Demand functions with limited budgets".
15074:
Convex analysis and minimization algorithms, Volume
14674:"Large deviations for sums of i.i.d. random compact sets"
13595:
12527:
that contains the original set. The Minkowski sum of two
10671:{\displaystyle X=\sum _{n\in I}X_{n}+\sum _{n\in J}q_{n}}
10130:{\displaystyle x=\sum _{n\in I}x_{n}+\sum _{n\in J}q_{n}}
15358:(first ed.). Palgrave Macmillan. pp. 653–661.
15067:
14712:(March 1991). "The Mathematization of economic theory".
14472:
Constrained optimization and Lagrange multiplier methods
13942:
convexity of sets derived by that averaging process
13731:
12267:. The Shapley–Folkman lemma enables a refinement of the
11292:
An optimal basket of goods occurs where the budget-line
9126:
Then, it is natural to consider the "variance" of a set
7259:{\displaystyle \{{\bar {q}}_{n,k}\}_{n\in 1:N,k\in 1:K}}
4867:{\displaystyle d^{2}(,\{0,1,2\})=1/4=d(,\{0,1,2\})^{2}.}
3929:
15384:
14448:(1999). "5.1.6 Separable problems and their geometry".
14146:"A strong law of large numbers for random compact sets"
13533:
13368:, pp. 1–126, especially 9–16 , 23–35 , and 35–37 )
11893:
summand-functions, each of which has its own argument:
11765:
11659:
8163:
since they are zero. The remaining terms take the form
4490:{\displaystyle d^{2}(x,S)=(\inf _{y\in S}\|x-y\|)^{2},}
2271:
15630:. Cambridge, Mass.: MIT Press. pp. 107–125.
15200:. New York: McGraw–Hill Book Company. pp. 1–126.
13574:, pp. xviii, 306–310, 312, 328–329, 347, and 352)
9480:{\displaystyle d(S,\mathrm {Conv} (S))^{2}\leq Var(S)}
7837:. At the same time, from the lifted representation of
90:
Related results provide more refined statements about
12004:
11928:
10983:
10957:
10713:
10684:
10606:
10579:
10550:
10474:
10423:
10396:
10369:
10342:
10316:
10169:
10143:
10065:
10016:
9936:
9878:
9753:
9733:
9692:
9672:
9637:
9617:
9597:
9551:
9495:
9410:
9285:
9152:
9132:
9092:
9034:
9014:
8994:
8961:
8915:
8866:
8826:
8792:
8721:
8701:
8681:
8575:
8534:
8502:
8482:
8446:
8420:
8384:
8364:
8299:
8169:
8136:
8023:
7872:
7843:
7823:
7794:
7755:
7700:
7657:
7637:
7598:
7575:
7443:
7415:
7373:
7277:
7187:
7158:
7062:
7037:
7008:
6981:
6849:
6814:
6785:
6710:
6658:
6619:
6599:
6516:
6456:
6436:
6400:
6361:
6332:
6312:
6082:
6055:
6028:
5995:
5966:
5902:
5794:
5750:
5700:
5672:
5652:
5621:
5575:
5511:
5491:
5471:
5425:
5405:
5370:
5341:
5303:
5273:
5151:
5050:
5011:
4970:
4903:
4883:
4733:
4648:
4576:
4503:
4411:
4301:
4252:
4239:{\displaystyle d^{2}(x,S)=\inf _{y\in S}\|x-y\|^{2}.}
4166:
4121:
4086:
4039:
4019:
3987:
3967:
3947:
3707:
3681:
3643:
3603:
3571:
3511:
3414:
3371:
3336:
3304:
3160:
3116:
3043:
3023:
2969:
2942:
2896:
2828:
2768:
2722:
2628:
2580:
2560:
2504:
2482:
2453:
2401:
2370:
2350:
2286:
2242:
2209:
2181:
2056:
1949:
1922:
1890:
1744:
1735:
By the principle of induction it is easy to see that
1645:
1514:
1437:
1405:
1379:
1310:
1258:
1206:
1090:
The definition of a convex set implies that the
871:
850:
824:
740:
connecting any two of its points is a subset of
658:
636:
611:
286:
233:
212:
186:
805:
if, for each pair of its points, every point on the
16105:(1943b). "A synthesis of pure demand analysis
15591:"The Brunn–Minkowski inequality and nonconvex sets"
14822:estimate in convex programming". In Ekeland, Ivar;
14008:
Optima: Mathematical Programming Society Newsletter
11564:has been emphasized by these laureates, along with
11146:has been emphasized by these laureates, along with
6586:{\displaystyle q_{n}=\sum _{k=1}^{K}w_{n,k}q_{n,k}}
6275:, Schneider, etc. An abstract and elegant proof by
5608:{\displaystyle B'\subset B\subset \mathbb {R} ^{D}}
258:
is the smallest convex set that contains
16004:
15072:(1993). "XII Abstract duality for practitioners".
14536:
12928:
12148:on problems that were known to be non-convex; for
12010:
11934:
11045:
10969:
10941:
10697:
10670:
10592:
10562:
10536:
10460:
10409:
10382:
10355:
10328:
10299:
10155:
10129:
10051:
10000:
9914:
9863:{\displaystyle Var=Var\leq E\leq rad(S)+\epsilon }
9862:
9739:
9719:
9678:
9658:
9623:
9603:
9583:
9537:
9479:
9390:
9271:
9138:
9115:
9078:
9020:
9000:
8980:
8947:
8901:
8852:
8805:
8778:
8707:
8687:
8667:
8556:
8520:
8488:
8468:
8432:
8406:
8370:
8347:
8285:
8155:
8122:
8010:
7858:
7829:
7809:
7780:
7741:
7686:
7643:
7623:
7581:
7561:
7427:
7401:
7359:
7258:
7173:
7144:
7043:
7023:
6994:
6967:
6835:
6800:
6772:{\displaystyle x=\sum _{n}\sum _{k}w_{n,k}q_{n,k}}
6771:
6693:
6644:
6605:
6585:
6502:
6442:
6415:
6382:
6347:
6318:
6247:
6068:
6041:
6014:
5981:
5952:
5884:
5771:
5736:
5683:
5658:
5638:
5607:
5549:
5497:
5477:
5457:
5411:
5391:
5356:
5309:
5289:
5256:
5128:
5032:
4994:
4954:
4889:
4866:
4717:
4634:
4562:
4489:
4395:
4287:
4238:
4145:
4107:
4065:
4025:
3993:
3981:, then its dimension is finite, and exactly
3973:
3953:
3918:
3687:
3667:
3629:
3589:
3557:
3494:
3397:
3354:
3322:
3290:
3142:
3102:
3029:
3009:
2955:
2928:
2868:
2814:
2754:
2708:
2606:
2566:
2544:
2488:
2468:
2439:
2385:
2356:
2334:
2260:
2228:
2195:
2164:
2036:
1928:
1908:
1868:
1727:
1628:
1497:
1423:
1385:
1361:
1296:
1244:
1153:) is the intersection of all the convex sets that
877:
856:
830:
768:joining two of its points is not a member of
664:
642:
617:
403:
239:
218:
192:
14318:Journal of Mathematical Analysis and Applications
13672:, pp. 93–94, 143, 318–319, 375–377, and 416)
12172:
11872:
8779:{\displaystyle \sum _{k}w'_{n,k}{\bar {q}}_{n,k}}
4246:And more generally, for any two nonempty subsets
1879:
1015:} of a vector space is any weighted average
17645:
16210:General equilibrium theory: An introduction
15909:
15407:Molchanov, Ilya (2005). "3 Minkowski addition".
15015:. Cambridge, Mass.: MIT Press. pp. 99–143.
14855:Competitive equilibrium: Theory and applications
14679:Proceedings of the American Mathematical Society
14205:Mathematical methods of game and economic theory
13556:, pp. 52–55, 145–146, 152–153, and 274–275)
13465:
13424:, pp. 1–2) uses results from Aumann (
12964:
12962:
12960:
12758:
12756:
12724:, pp. 364–381) describes an application of
12494:Milton's description of concavity serves as the
11046:{\displaystyle Var\leq \sum _{\max D}Var(Q_{n})}
11010:
10894:
9965:
9211:
9175:
8414:, together with the fact that there are at most
5848:
5327:
5279:
5211:
5096:
5073:
4599:
4444:
4336:
4196:
3675:. Note that the reindexing depends on the point
1498:{\displaystyle A+B:=\{x+y\mid x\in A,~y\in B\}.}
498:. Two points in the Cartesian plane can be
276:. For example, adding the set consisting of the
25:The Shapley–Folkman lemma is illustrated by the
14774:"12 Duality approaches to microeconomic theory"
14144:Artstein, Zvi; Vitale, Richard A. (1975).
13154:"A simple proof of the Shapley-Folkman theorem"
11445:were collected in an annotated bibliography by
11066:A Winner of the 2012 Nobel Award in Economics,
10052:{\displaystyle \forall x\in \mathrm {Conv} (Q)}
8902:{\displaystyle \forall x\in \mathrm {Conv} (S)}
6503:{\displaystyle q_{n}\in \mathrm {Conv} (Q_{n})}
5896:In particular, if we have an infinite sequence
4288:{\displaystyle S,S'\subseteq \mathbb {R} ^{D},}
3558:{\displaystyle q_{n}\in \mathrm {Conv} (Q_{n})}
2815:{\displaystyle q_{n}\in \mathrm {Conv} (Q_{n})}
1116:of the red set, each blue point is a
109:between any point in the Minkowski sum and its
15723:. Probability and its applications. Springer.
15693:Convex bodies: The Brunn–Minkowski theory
15644:Microéconomie: Les défaillances du marché
15304:The Theory of general economic equilibrium: A
14143:
13837:, pp. 195–198, 218, 232, 237–238 and 407)
13692:
13690:
12762:
11383:). The main contributors were Farrell, Bator,
6258:
5615:be two nested balls, then the circumradius of
127:The Shapley–Folkman lemma has applications in
17216:
16549:
16300:
15753:
15718:
15626:Salanié, Bernard (2000). "7 Nonconvexities".
15198:Three essays on the state of economic science
13810:
13461:
13459:
13406:
12957:
12753:
12704:
12702:
11652:whose convergence proofs are stated for only
11411:Starr's 1969 paper and contemporary economics
6290:of the representation, but do not provide an
5550:{\displaystyle x\in \mathrm {Conv} (S\cap B)}
4635:{\displaystyle d(x,S)=\inf _{y\in S}\|x-y\|.}
1165:. For example, the convex hull of the set of
16011:(3rd ed.). W. W. Norton & Company.
14732:
14229:
13938:then the resulting set is necessarily convex
13798:
13786:
13780:
13743:
13741:
13388:
12716:
12599:problem—that is, the problem defined as the
12558:
11611:
11273:passes one indifference curve. A consumer's
10200:
10170:
9821:
9808:
7217:
7188:
5123:
5111:
4955:{\displaystyle d^{2}(\mathrm {Conv} (Q),Q).}
4848:
4830:
4783:
4765:
4626:
4614:
4471:
4459:
4381:
4368:
4224:
4211:
3798:
3777:
3317:
3305:
3282:
3270:
3244:
3232:
2329:
2299:
1860:
1779:
1623:
1605:
1599:
1551:
1545:
1533:
1527:
1515:
1489:
1450:
1431:is defined to be the element-wise operation
395:
377:
371:
323:
317:
305:
299:
287:
147:. In probability, it can be used to prove a
15885:General equilibrium theory: An introduction
15486:
15449:
15355:The new Palgrave: A dictionary of economics
15147:Cowles Foundation for Research in Economics
15037:
14968:
14119:. In Hart, Sergiu; Neyman, Abraham (eds.).
13936:over a collection of insignificant agents,
13877:
13849:
13845:
13843:
13708:
13687:
13559:
13511:
13493:
13189:
13187:
12872:
12860:
12844:
12828:
12824:
12808:
12740:"), where non-convexity appears because of
12685:
12683:
12681:
12679:
12677:
12546:
11277:(relative to an indifference curve) is the
7054:With this, we have a lifted representation
4108:{\displaystyle S\subseteq \mathbb {R} ^{D}}
76:, but was first published by the economist
17223:
17209:
16556:
16542:
16307:
16293:
16141:
15556:
15527:
15332:
15298:
15266:
15246:
13975:
13720:
13553:
13517:
13456:
13446:
13382:
13378:
13371:
13357:
13261:
13205:
12777:
12775:
12773:
12771:
12699:
11819:A real-valued function is defined to be a
8358:Combining the fact that for each value of
8130:We drop all terms on the r.h.s. for which
5044:(as shown in the diagram). More formally,
4995:{\displaystyle S\subset \mathbb {R} ^{D},}
4008:
16270:New Palgrave Dictionary of Economics
16084:
15689:
15649:
15642:English translation of the (1998) French
15608:
15406:
15102:
15007:
14852:
14691:
14466:
14444:
14329:
14161:
14026:
13969:
13834:
13738:
13696:
13571:
13546:
13505:
13477:
13417:
13415:
13394:
13273:
13245:
13221:
13209:
13193:
13085:
13019:
12992:
12980:
12951:
12924:
12900:
12888:
12882:
12856:
12840:
12820:
12793:
12721:
12708:
12665:
12663:
12661:
12541:the inclusion can be strict even for two
12499:
12464:"Eternal darkness" describes the Hell of
11487:For each quasi-equilibrium's prices
11310:
11090:, who was investigating the existence of
10425:
9216:
9100:
7011:
6817:
6788:
6403:
6364:
6335:
6301:
5969:
5779:, the following theorem is a refinement:
5772:{\displaystyle S\subset \mathbb {R} ^{D}}
5759:
5595:
5392:{\displaystyle S\subset \mathbb {R} ^{D}}
5379:
4979:
4272:
4130:
4095:
2456:
2373:
2189:
486:in which every point is identified by an
16:Sums of sets of vectors are nearly convex
16169:
16147:Demand analysis: A study in econometrics
15946:SIAM Journal on Control and Optimization
15943:
15853:The new Palgrave dictionary of economics
15719:Schneider, Rolf; Weil, Wolfgang (2008).
15214:
15188:
15161:
14939:
14910:
14828:Convex analysis and variational problems
14064:
13963:
13957:
13901:
13895:
13840:
13653:
13651:
13630:
13365:
13349:
13341:
13312:
13308:
13184:
13055:
13043:
12804:
12802:
12674:
12565:, pp. 362–364) also considered the
12216:-valued random vectors equals
11764:
11667:relies on the following definitions for
11615:
11414:
11314:
11289:if all such preference sets are convex.
11185:
11122:: Shapley himself (2012), Arrow (1972),
11061:
8348:{\displaystyle 1=\sum _{k}w'_{n_{0},k}.}
5666:, but its inner radius is the radius of
5559:
4718:{\displaystyle d^{2}(S,S')=d(S,S')^{2}.}
2617:
1192:
1107:
20:
17313:Locally convex topological vector space
15972:
15625:
14879:
14814:
14771:
14645:(1981). "Appendix A Convex sets".
14641:
14609:
13822:
13789:, pp. 226, 233, 235, 238, and 241)
13756:. Séries A et B (in French).
13747:
13681:
13642:
13618:
13584:
13450:
13345:
13344:, p. 478) and others—for example,
13304:
13291:
13285:
13031:
13007:
12768:
12712:
12689:
12562:
12441:. Lyapunov's theorem has been called a
12291:is convex. Here, the traditional term "
9584:{\displaystyle x\in \mathrm {Conv} (S)}
5458:{\displaystyle x\in \mathrm {Conv} (S)}
5040:to be the infimum of the radius of all
2929:{\displaystyle x\in \mathrm {Conv} (Q)}
2755:{\displaystyle x\in \mathrm {Conv} (Q)}
158:
113:. This upper bound is sharpened by the
94:the approximation is. For example, the
17646:
15999:
15043:Core and equilibria of a large economy
14708:
14568:
14542:Nanta Mathematica (Nanyang University)
14510:IEEE Transactions on Automatic Control
14415:
14386:
14346:
14308:
14268:
14042: ed.). Amsterdam: North-Holland.
13991:
13925:
13748:Ekeland, Ivar (1974). "Une estimation
13734:, pp. 143–145, 151, 153, and 156)
13725:
13702:
13669:
13663:
13534:Mas-Colell, Whinston & Green (1995
13528:
13429:
13425:
13421:
13412:
13353:
13329:
13325:
12927:, p. 140) credits this result to
12781:
12658:
12648:
12646:
12341:is a vector measure, where
12181:. Each point in the convex hull of a (
11070:proved the Shapley–Folkman lemma with
9927:Proof of Shapley–Folkman–Starr theorem
7742:{\displaystyle ({\bar {x}})_{D+n_{0}}}
7631:, there must be at least one value of
7435:of them are nonzero. Since we defined
6392:Carathéodory's theorem for conic hulls
4563:{\displaystyle d^{2}(x,S)=d(x,S)^{2},}
4146:{\displaystyle x\in \mathbb {R} ^{D},}
3698:The lemma may be stated succinctly as
3010:{\displaystyle \sum _{n=1}^{N}q_{n}=x}
2869:{\displaystyle \sum _{n=1}^{N}q_{n}=x}
2552:is the Minkowski sum of the summands.
2545:{\displaystyle Q=\sum _{n=1}^{N}Q_{n}}
2364:is the dimension of the ambient space
17204:
16537:
16288:
16257:
16193:
16101:
16073:Probability Theory and Related Fields
15878:
15837:
15805:
15585:
15521:) Princeton Mathematical Series
14572:Foundations of mathematical economics
14474:. Belmont, Mass.: Athena Scientific.
14202:
13934:if one averages those individual sets
13889:
13732:Hiriart-Urruty & Lemaréchal (1993
13657:
13648:
13565:
13562:, pp. 37, 115–116, 122, and 168)
13481:
13442:
13361:
13257:
13233:
13113:
13093:. Belmont, Mass.: Athena Scientific.
13056:Anderson, Robert M. (14 March 2005).
13003:
13001:
12968:
12939:
12912:
12876:
12799:
12669:
12638:
12634:
12632:
12630:
12628:
12626:
12531:need not be closed, so the following
12503:
12489:Betwixt Damiata and Mount Casius old,
12487:A gulf profound as that Serbonian Bog
12474:, whose concavity is compared to the
12189:of a finite-dimensional space is the
12073:of points in the convexified problem
11086:was first published by the economist
7694:is nonzero. Remember that we defined
3930:The converse of Shapley–Folkman lemma
1188:
466:
16505:List of differential geometry topics
16062:
16033:
15134:
15078:: Advanced theory and bundle methods
14671:
13913:
13873:
13861:
13752:en programmation non convexe".
13151:
12652:
12425:Lyapunov's theorem has been used in
11660:Preliminaries of optimization theory
11580:, in general-equilibrium theory, in
10537:{\displaystyle Var<Var+\epsilon }
9243: is finitely supported in
8528:for which there are at least two of
3961:, and for no number less than
2335:{\displaystyle D,N\in \{1,2,3,...\}}
2272:Statements of the three main results
978:is convex if and only if every
707:multiplication by a real number
16003:(1992). "21.2 Convexity and size".
14973:; Intriligator, Michael D. (eds.).
14780:; Intriligator, Michael D. (eds.).
14314:"Integrals of set-valued functions"
12643:
12226:probabilistic theory of random sets
12177:Convex sets are often studied with
11877:In many optimization problems, the
11560:(1970); the complementary topic of
11142:(1970); the complementary topic of
10705:is a deterministic vector, we have
6022:such that the inner radius of each
5953:{\displaystyle (Q_{n})_{n=1,2,...}}
2476:. They are also called "summands".
1100:is the empty set, which is convex.
1008:, . . . ,
897:is convex, either by definition or
175:in the set: For example, the solid
68:. It is named after mathematicians
29:of four sets. The point (+) in the
13:
17112:Microfoundations of macroeconomics
16563:
16170:Anderson, Robert M. (March 2005).
14975:Handbook of mathematical economics
14782:Handbook of mathematical economics
14736:Mathematics of Operations Research
14233:Mathematics of Operations Research
12998:
12711:, pp. 364–381) acknowledging
12623:
12545:closed summand-sets, according to
12141:was surprised by his success with
11248:which is preferred but unfeasible.
10461:{\displaystyle \mathbb {E} =x_{n}}
10251:
10248:
10245:
10242:
10036:
10033:
10030:
10027:
10017:
9568:
9565:
9562:
9559:
9433:
9430:
9427:
9424:
9308:
9305:
9302:
9299:
9195:
9192:
9189:
9186:
8886:
8883:
8880:
8877:
8867:
8853:{\displaystyle \mathrm {Conv} (S)}
8837:
8834:
8831:
8828:
7051:, and 0 at all other coordinates.
6836:{\displaystyle \mathbb {R} ^{D+N}}
6694:{\displaystyle \sum _{k}w_{n,k}=1}
6480:
6477:
6474:
6471:
6383:{\displaystyle \mathbb {R} ^{D+N}}
6108:
6105:
6102:
6099:
5824:
5821:
5818:
5815:
5528:
5525:
5522:
5519:
5442:
5439:
5436:
5433:
5181:
5178:
5175:
5172:
4927:
4924:
4921:
4918:
4066:{\displaystyle \mathrm {Conv} (Q)}
4050:
4047:
4044:
4041:
3859:
3856:
3853:
3850:
3718:
3715:
3712:
3709:
3535:
3532:
3529:
3526:
3398:{\displaystyle \mathrm {Conv} (Q)}
3382:
3379:
3376:
3373:
3263:
3260:
3257:
3254:
3225:
3222:
3219:
3216:
3067:
3064:
3061:
3058:
2913:
2910:
2907:
2904:
2792:
2789:
2786:
2783:
2739:
2736:
2733:
2730:
2686:
2683:
2680:
2677:
2639:
2636:
2633:
2630:
2607:{\displaystyle \mathrm {Conv} (Q)}
2591:
2588:
2585:
2582:
2142:
2139:
2136:
2133:
2067:
2064:
2061:
2058:
2018:
2015:
2012:
2009:
1992:
1989:
1986:
1983:
1960:
1957:
1954:
1951:
1185:, which contains the unit circle.
908:is convex if, for all points
432:However, the distance between the
14:
17705:
16163:
15628:Microeconomics of market failures
15378:PDF file at Mas-Colell's homepage
15272:Journal of Mathematical Economics
15164:Game theory for economic analysis
14494:Reprint of (1982) Academic Press.
12275:, which is defined on subsets of
12246:probabilistic limit theorems
12208:. Thus, for a non-empty set
11497:At quasi-equilibrium prices
10951:Since this is true for arbitrary
9538:{\displaystyle Var(S)\leq rad(S)}
9079:{\displaystyle Pr(X=q_{n})=w_{n}}
8948:{\displaystyle x=\sum w_{n}q_{n}}
5625:
3936:converse of Shapley–Folkman lemma
3087:
2447:are nonempty, bounded subsets of
2196:{\displaystyle N\in \mathbb {N} }
2047:And by induction it follows that
1128:of a real vector space, its
809:that joins them is still in
33:of the Minkowski sum of the four
17185:
17184:
17173:
16518:
16517:
15721:Stochastic and integral geometry
15530:The Journal of Political Economy
15392:. Oxford University Press.
15302:(1985). "1.L Averages of sets".
15220:The Journal of Political Economy
14883:The Journal of Political Economy
14389:The Journal of Political Economy
13591:Fundamentals of public economics
13260:, pp. 231 and 239–240) and
12696:'s experiments on page 373.
12480:Book II, lines 592–594
11865:(pictured) is non-convex on the
11825:if its epigraph is a convex set.
11376:The Journal of Political Economy
11329:However, if a preference set is
11098:. In his paper, Starr studied a
9720:{\displaystyle rad(S)+\epsilon }
8988:, we can define a random vector
8567:Thus we obtain a representation
7024:{\displaystyle \mathbb {R} ^{N}}
6801:{\displaystyle \mathbb {R} ^{D}}
6416:{\displaystyle \mathbb {R} ^{D}}
6348:{\displaystyle \mathbb {R} ^{D}}
5982:{\displaystyle \mathbb {R} ^{D}}
5960:of nonempty, bounded subsets of
2469:{\displaystyle \mathbb {R} ^{D}}
2386:{\displaystyle \mathbb {R} ^{D}}
2229:{\displaystyle Q_{n}\subseteq X}
1362:{\displaystyle Q_{1}+Q_{2}=^{2}}
1200:of sets. The sum of the squares
994:of an indexed subset {
749:
721:
454:
17418:Ekeland's variational principle
16172:"1 The Shapley–Folkman theorem"
15068:Hiriart-Urruty, Jean-Baptiste;
14575:. Cambridge, Mass.: MIT Press.
13907:
13883:
13867:
13855:
13828:
13816:
13804:
13792:
13714:
13675:
13636:
13624:
13621:, pp. 112–113 and 107–115)
13612:
13578:
13540:
13522:
13499:
13487:
13471:
13435:
13400:
13335:
13318:
13297:
13267:
13251:
13239:
13227:
13215:
13199:
13145:
13107:
13079:
13058:"1 The Shapley–Folkman theorem"
13049:
13037:
13025:
13013:
12986:
12974:
12945:
12933:
12918:
12906:
12894:
12866:
12850:
12834:
12814:
12552:
12509:
12458:
12133:This analysis was published by
11483:with the following properties:
11161:
9915:{\displaystyle Var(S)\leq r(S)}
9686:is bounded in a ball of radius
7495:
6901:
5737:{\displaystyle r(S)\leq rad(S)}
3668:{\displaystyle D+1\leq n\leq N}
2440:{\displaystyle Q_{1},...,Q_{N}}
1064:} satisfying the equation
926:and for every real number
247:is not a subset of the circle.
171:joining two of its points is a
16127:10.1080/03461238.1943.10404737
16115:Scandinavian Actuarial Journal
16111:Skandinavisk Aktuarietidskrift
14857:. Cambridge University Press.
14818:(1999) . "Appendix I: An
14001:interview - Claude Lemaréchal"
12787:
12200:that takes its values in
12173:Probability and measure theory
11873:Additive optimization problems
11693:and function evaluations
11449:. He gave the bibliography to
11391:-papers stimulated a paper by
11040:
11027:
10999:
10993:
10970:{\displaystyle \epsilon >0}
10924:
10911:
10872:
10859:
10729:
10723:
10570:is an arbitrary small number.
10563:{\displaystyle \epsilon >0}
10525:
10512:
10497:
10484:
10442:
10429:
10268:
10255:
10215:
10207:
10046:
10040:
9995:
9982:
9909:
9903:
9894:
9888:
9851:
9845:
9830:
9805:
9796:
9784:
9769:
9763:
9708:
9702:
9647:
9641:
9578:
9572:
9532:
9526:
9511:
9505:
9474:
9468:
9447:
9443:
9437:
9414:
9385:
9379:
9364:
9358:
9349:
9343:
9322:
9318:
9312:
9289:
9266:
9260:
9226:
9220:
9205:
9199:
9168:
9162:
9116:{\displaystyle x=\mathbb {E} }
9110:
9104:
9060:
9041:
8896:
8890:
8847:
8841:
8758:
8642:
8582:
8232:
8206:
8196:
8095:
8076:
8066:
7966:
7889:
7882:
7873:
7850:
7801:
7775:
7756:
7717:
7710:
7701:
7556:
7524:
7503:
7489:
7459:
7450:
7402:{\displaystyle w'_{n,k}\geq 0}
7339:
7284:
7198:
7165:
7121:
7069:
6962:
6930:
6909:
6895:
6865:
6856:
6704:Now "lift" the representation
6497:
6484:
6427:Proof of Shapley–Folkman lemma
5917:
5903:
5873:
5859:
5837:
5834:
5828:
5805:
5731:
5725:
5710:
5704:
5544:
5532:
5452:
5446:
5351:
5345:
5290:{\displaystyle \sum _{\max D}}
5242:
5228:
5194:
5191:
5185:
5162:
5066:
5060:
5027:
5021:
4946:
4937:
4931:
4914:
4852:
4824:
4812:
4809:
4786:
4759:
4747:
4744:
4703:
4685:
4676:
4659:
4592:
4580:
4548:
4535:
4526:
4514:
4475:
4440:
4434:
4422:
4329:
4312:
4189:
4177:
4060:
4054:
3876:
3863:
3816:
3808:
3630:{\displaystyle q_{n}\in Q_{n}}
3552:
3539:
3392:
3386:
3349:
3337:
3285:
3267:
3247:
3229:
3209:
3197:
3191:
3179:
3173:
3161:
3143:{\displaystyle q_{n}\in Q_{n}}
3084:
3071:
2923:
2917:
2809:
2796:
2749:
2743:
2703:
2690:
2649:
2643:
2601:
2595:
2159:
2146:
2105:
2071:
2028:
2022:
2002:
1996:
1976:
1964:
1909:{\displaystyle A,B\subseteq X}
1880:Convex hulls of Minkowski sums
1424:{\displaystyle A,B\subseteq X}
1350:
1337:
1285:
1272:
1233:
1220:
1145:convex set that contains
1103:
712:
574:by each real number
1:
16036:International Economic Review
14983:10.1016/S1573-4382(81)01005-9
14790:10.1016/S1573-4382(82)02007-4
14693:10.1090/S0002-9939-99-04788-7
13984:
13588:(1988). "3. Nonconvexities".
13466:Starr & Stinchcombe (1999
13449:, p. 146), according to
12738:unit commitment problems
12491:Where Armies whole have sunk.
11194:every basket of goods on the
6645:{\displaystyle w_{n,k}\geq 0}
5784:Shapley–Folkman–Starr theorem
5639:{\displaystyle B\setminus B'}
5328:Shapley–Folkman–Starr theorem
3590:{\displaystyle 1\leq n\leq D}
2261:{\displaystyle 1\leq n\leq N}
115:Shapley–Folkman–Starr theorem
17674:Geometric transversal theory
16314:
15559:Journal of Political Economy
15488:Rockafellar, R. Tyrrell
15284:10.1016/0304-4068(78)90010-1
14942:Journal of Political Economy
14913:Journal of Political Economy
14772:Diewert, W. E. (1982).
14714:The American Economic Review
14647:Economics for mathematicians
14418:Journal of Political Economy
14331:10.1016/0022-247X(65)90049-1
14036:General competitive analysis
13948:in the commodity space
13864:, pp. 203, and 205–206)
13131:10.1016/0022-0531(81)90010-7
12929:Borwein & O'Brien (1978)
12152:problems, a solution of the
12026:with the minimum value
11534:non-convex sets in economics
11181:
11116:non-convex sets in economics
9487:: Expand their definitions.
7687:{\displaystyle w'_{n_{0},k}}
7624:{\displaystyle n_{0}\in 1:N}
3298:is the sum of an element in
3154:For example, every point in
7:
17438:Hermite–Hadamard inequality
17057:Civil engineering economics
17042:Statistical decision theory
16682:Income elasticity of demand
14643:Cassels, J. W. S.
13364:, p. 26)—commented on
12763:Artstein & Vitale (1975
12521:closure of the original set
12498:prefacing chapter seven of
11685:is the set of the pairs of
11648:can be nearly solved (with
11443:non-convexity and economics
10336:, construct random vectors
9922:: use the previous result.
8156:{\displaystyle n\neq n_{0}}
6259:Other proofs of the results
6015:{\displaystyle r_{0}\geq 0}
5989:, and if there exists some
5324:on the number of summands.
2496:is the number of summands.
2276:
1079:+ . . . +
901:, depending on the author.
484:Cartesian coordinate system
272:is the addition of the set
10:
17710:
17679:General equilibrium theory
16692:Price elasticity of supply
16687:Price elasticity of demand
16677:Cross elasticity of demand
15862:10.1057/9780230226203.1518
15364:10.1057/9780230226203.3173
15216:Koopmans, Tjalling C.
15190:Koopmans, Tjalling C.
15162:Ichiishi, Tatsuro (1983).
14468:Bertsekas, Dimitri P.
14446:Bertsekas, Dimitri P.
13811:Schneider & Weil (2008
13407:Shapley & Shubik (1966
13119:Journal of Economic Theory
13091:Convex Optimization Theory
12717:Aubin & Ekeland (1976)
12559:Aubin & Ekeland (1976)
12269:Brunn–Minkowski inequality
12228:, for example, to prove a
11322:
11251:
11226:), where the budget line (
11205:over each basket on
11057:
8981:{\displaystyle q_{n}\in S}
8293:, so we find the equation
7859:{\displaystyle {\bar {x}}}
7810:{\displaystyle {\bar {x}}}
7174:{\displaystyle {\bar {x}}}
6613:is a large finite number,
5267:where we use the notation
4155:squared Euclidean distance
2574:is an arbitrary vector in
1297:{\displaystyle Q_{2}=^{2}}
1245:{\displaystyle Q_{1}=^{2}}
1169: {0,1} is the closed
904:More formally, a set
893:shape, is non-convex. The
791:In a real vector space, a
697: } on which two
17623:
17590:
17545:
17476:
17402:
17326:
17268:
17242:
17168:
17135:
17014:
16571:
16513:
16480:
16449:
16399:
16347:
16322:
16263:"Shapley–Folkman theorem"
15840:"Shapley–Folkman theorem"
15729:10.1007/978-3-540-78859-1
15472:10.1017/S0305004100062691
15194:Koopmans, Tjalling C
14853:Ellickson, Bryan (1994).
14627:10.1017/S0305004100051884
14150:The Annals of Probability
13787:Aubin & Ekeland (1976
13645:, pp. 127 and 33–34)
12230:law of large numbers
11973:For example, problems of
11612:Mathematical optimization
11536:has been studied by many
11441:Previous publications on
11325:Non-convexity (economics)
11172:mathematical optimization
11118:has been studied by many
10600:be independent. Then let
10390:is finitely supported on
9611:be finitely supported in
7781:{\displaystyle (D+n_{0})}
7031:that has 1 at coordinate
781:test whether a subset be
688:, . . . ,
17624:Applications and related
17428:Fenchel-Young inequality
16748:Income–consumption curve
16218:10.1017/CBO9781139174749
15973:Trockel, Walter (1984).
15927:10.1017/CBO9780511896583
15690:Schneider, Rolf (1993).
14863:10.1017/CBO9780511609411
14569:Carter, Michael (2001).
14522:10.1109/tac.1983.1103136
14109:Nobel Prize in Economics
13918:Nobel Prize in Economics
13878:Puri & Ralescu (1985
13850:Puri & Ralescu (1985
13494:Green & Heller (1981
13480:, pp. 169–182) and
13348:, pp. 390–391) and
12829:Green & Heller (1981
12595:closure of a non-convex
12523:, which is the smallest
12452:
12283:, which states that the
12158:constraint qualification
11628:if the region above its
11562:convex sets in economics
11144:convex sets in economics
9008:, finitely supported in
8860:in probabilistic terms:
8557:{\displaystyle w_{n,k}'}
8521:{\displaystyle n\in 1:N}
8469:{\displaystyle w_{n,k}'}
8407:{\displaystyle w_{n,k}'}
7181:is in the conic hull of
5744:for any bounded subset
5297:to mean "the sum of the
4080:For any nonempty subset
2880:refines this statement.
1916:of a real vector space,
1505:(See also.) For example
1120:of some red points.
878:{\displaystyle \oslash }
831:{\displaystyle \bullet }
568:further, a point can be
262:. This distance is zero
240:{\displaystyle \oslash }
201:is a convex set but the
193:{\displaystyle \bullet }
17384:Legendre transformation
17308:Legendre transformation
17082:Industrial organization
16500:List of geometry topics
16063:Weil, Wolfgang (1982).
15844:Durlauf, Steven N.
15838:Starr, Ross M. (2008).
15651:Samuelson, Paul A.
15610:10.1023/A:1004958110076
15514:. Reprint of the 1970 (
13950:obtained by aggregation
13799:Di Guglielmo (1977
13598:Press. pp. 63–65.
13447:Wold & Juréen (1953
13262:Wold & Juréen (1953
13152:Zhou, Lin (June 1993).
12827:, pp. 10–11), and
12748:Bertsekas et al. (1983)
12734:electrical power plants
12449:of Lyapunov's theorem.
11831:quadratic function
11519:Starr established that
5141:Shapley–Folkman theorem
4964:For any bounded subset
4497:so we can simply write
4009:Shapley–Folkman theorem
3323:{\displaystyle \{0,1\}}
2936:, there exist elements
2762:, there exist elements
2342:are positive integers.
813:. For example, a solid
58:Minkowski addition
17654:Additive combinatorics
17631:Convexity in economics
17565:(lower) ideally convex
17423:Fenchel–Moreau theorem
17413:Carathéodory's theorem
16007:Microeconomic Analysis
15915:Chichilnisky, Graciela
14672:Cerf, Raphaël (1999).
14348:Aumann, Robert J.
14310:Aumann, Robert J.
14270:Aumann, Robert J.
14163:10.1214/aop/1176996275
14115:Artstein, Zvi (1995).
14028:Arrow, Kenneth J.
13954:
13547:Arrow & Hahn (1980
13478:Arrow & Hahn (1980
13445:, p. 243) and by
13395:Arrow & Hahn (1980
13282:
13222:Arrow & Hahn (1980
13210:Arrow & Hahn (1980
13208:, pp. 58–61) and
12993:Arrow & Hahn (1980
12901:Arrow & Hahn (1980
12841:Arrow & Hahn (1980
12821:Arrow & Hahn (1980
12794:Arrow & Hahn (1980
12500:Arrow & Hahn (1980
12493:
12204:, as a consequence of
12012:
11936:
11778:
11665:Nonlinear optimization
11637:
11594:mathematical economics
11525:
11438:
11360:
11320:
11311:Non-convex preferences
11254:Convexity in economics
11249:
11075:
11047:
10971:
10943:
10699:
10672:
10594:
10564:
10538:
10462:
10411:
10384:
10357:
10330:
10329:{\displaystyle n\in I}
10301:
10157:
10131:
10053:
10002:
9916:
9864:
9741:
9721:
9680:
9660:
9625:
9605:
9585:
9539:
9481:
9392:
9273:
9140:
9117:
9080:
9022:
9002:
8982:
8949:
8903:
8854:
8807:
8780:
8709:
8689:
8669:
8558:
8522:
8490:
8470:
8434:
8408:
8372:
8349:
8287:
8157:
8124:
8012:
7860:
7831:
7811:
7782:
7743:
7688:
7645:
7625:
7592:We argue that for any
7583:
7563:
7429:
7403:
7361:
7260:
7175:
7146:
7045:
7025:
6996:
6969:
6837:
6802:
6773:
6695:
6646:
6607:
6587:
6550:
6504:
6444:
6417:
6384:
6349:
6320:
6302:A proof of the results
6249:
6196:
6147:
6070:
6043:
6016:
5983:
5954:
5886:
5773:
5738:
5685:
5660:
5640:
5609:
5566:
5551:
5499:
5479:
5465:, there exists a ball
5459:
5413:
5393:
5358:
5311:
5291:
5258:
5130:
5034:
5033:{\displaystyle rad(S)}
4996:
4956:
4891:
4868:
4719:
4636:
4564:
4491:
4397:
4289:
4240:
4147:
4109:
4067:
4027:
3995:
3975:
3955:
3920:
3747:
3689:
3669:
3631:
3591:
3559:
3496:
3481:
3441:
3399:
3356:
3324:
3292:
3144:
3104:
3031:
3011:
2990:
2957:
2930:
2870:
2849:
2816:
2756:
2710:
2675:
2608:
2568:
2546:
2531:
2490:
2470:
2441:
2387:
2358:
2336:
2262:
2230:
2203:and non-empty subsets
2197:
2166:
2131:
2094:
2038:
1930:
1910:
1870:
1802:
1765:
1729:
1666:
1630:
1499:
1425:
1399:of two non-empty sets
1387:
1370:
1363:
1298:
1246:
1124:For every subset
1121:
972:mathematical induction
879:
858:
857:{\displaystyle \circ }
832:
666:
644:
619:
447:average includes more
405:
241:
220:
219:{\displaystyle \circ }
194:
42:
17553:Convex series related
17453:Shapley–Folkman lemma
17052:Engineering economics
16647:Cost–benefit analysis
16472:Differential geometry
15801:on 24 September 2017.
15425:10.1007/1-84628-150-4
15411:Theory of random sets
15260:IRIA (now INRIA)
14971:Arrow, Kenneth Joseph
14778:Arrow, Kenneth Joseph
14605:on 15 September 2006.
14499:Bertsekas, Dimitri P.
14450:Nonlinear Programming
14107:, winner of the 2008
14105:Robert J. Aumann
13929:
13586:Laffont, Jean-Jacques
13381:, p. 447,
13356:, pp. 482–483),
13277:
13087:Bertsekas, Dimitri P.
12715:on page 374 and
12485:
12357:is defined for every
12319:measurable space
12234:central limit theorem
12013:
11937:
11845:is convex, as is the
11768:
11757:is the set of points
11619:
11521:
11418:
11355:
11318:
11189:
11065:
11048:
10972:
10944:
10700:
10698:{\displaystyle q_{n}}
10673:
10595:
10593:{\displaystyle X_{n}}
10565:
10539:
10463:
10412:
10410:{\displaystyle Q_{n}}
10385:
10383:{\displaystyle X_{n}}
10358:
10356:{\displaystyle X_{n}}
10331:
10302:
10158:
10132:
10054:
10003:
9917:
9865:
9742:
9722:
9681:
9661:
9626:
9606:
9586:
9540:
9482:
9393:
9274:
9141:
9118:
9081:
9023:
9003:
8983:
8950:
8904:
8855:
8808:
8806:{\displaystyle Q_{n}}
8781:
8710:
8690:
8670:
8559:
8523:
8491:
8471:
8435:
8409:
8373:
8350:
8288:
8158:
8125:
8013:
7861:
7832:
7812:
7783:
7744:
7689:
7646:
7626:
7584:
7564:
7430:
7404:
7362:
7261:
7176:
7147:
7046:
7026:
6997:
6995:{\displaystyle e_{n}}
6970:
6838:
6803:
6774:
6696:
6647:
6608:
6588:
6530:
6505:
6445:
6418:
6385:
6350:
6321:
6250:
6176:
6127:
6071:
6069:{\displaystyle r_{0}}
6044:
6042:{\displaystyle Q_{n}}
6017:
5984:
5955:
5887:
5774:
5739:
5686:
5661:
5641:
5610:
5563:
5552:
5500:
5480:
5460:
5414:
5399:to be the infimum of
5394:
5364:of a bounded subset
5359:
5312:
5292:
5259:
5131:
5035:
4997:
4957:
4897:is by upper-bounding
4892:
4869:
4720:
4637:
4565:
4492:
4398:
4290:
4241:
4148:
4110:
4068:
4028:
3996:
3976:
3956:
3921:
3727:
3690:
3670:
3632:
3592:
3560:
3497:
3455:
3421:
3405:can be decomposed as
3400:
3357:
3325:
3293:
3145:
3105:
3032:
3012:
2970:
2958:
2956:{\displaystyle q_{n}}
2931:
2885:Shapley–Folkman lemma
2878:Shapley–Folkman lemma
2871:
2829:
2817:
2757:
2711:
2655:
2618:Shapley–Folkman lemma
2609:
2569:
2547:
2511:
2491:
2471:
2442:
2388:
2359:
2337:
2263:
2231:
2198:
2167:
2111:
2074:
2039:
1931:
1911:
1871:
1782:
1745:
1730:
1646:
1631:
1500:
1426:
1388:
1364:
1299:
1247:
1196:
1111:
1042:+ . . . +
986:also belongs to
880:
859:
833:
667:
645:
625:can be viewed as the
620:
406:
242:
221:
195:
47:Shapley–Folkman
24:
17694:Theorems in geometry
17443:Krein–Milman theorem
17236:variational analysis
16869:Price discrimination
16763:Intertemporal choice
15390:Microeconomic theory
14749:10.1287/moor.2.3.285
14599:answers to exercises
14246:10.1287/moor.1.3.225
14196:euclid.ss/1176996275
13928:, p. 4) wrote:
13880:, pp. 154–155).
13360:, p. 438), and
13276:, pp. 359–360):
12317:defined on the same
12315:probability measures
12206:Carathéodory's lemma
12150:minimizing nonlinear
12002:
11926:
11751:real-valued function
11529:Roger Guesnerie
11109:Roger Guesnerie
11094:while studying with
10981:
10955:
10711:
10682:
10604:
10577:
10548:
10472:
10421:
10394:
10367:
10340:
10314:
10167:
10141:
10063:
10014:
9934:
9930:It suffices to show
9876:
9751:
9731:
9690:
9670:
9635:
9615:
9595:
9549:
9493:
9408:
9283:
9150:
9130:
9090:
9032:
9012:
8992:
8959:
8913:
8864:
8824:
8790:
8719:
8699:
8679:
8573:
8532:
8500:
8480:
8444:
8418:
8382:
8378:there is a non-zero
8362:
8297:
8167:
8134:
8021:
7870:
7841:
7821:
7792:
7753:
7698:
7655:
7635:
7596:
7573:
7441:
7413:
7371:
7275:
7185:
7156:
7060:
7035:
7006:
6979:
6847:
6812:
6783:
6708:
6656:
6617:
6597:
6514:
6454:
6434:
6398:
6394:, then drop back to
6359:
6330:
6310:
6080:
6053:
6049:is upper-bounded by
6026:
5993:
5964:
5900:
5792:
5748:
5698:
5670:
5650:
5619:
5573:
5509:
5489:
5469:
5423:
5403:
5368:
5357:{\displaystyle r(S)}
5339:
5301:
5271:
5149:
5048:
5009:
4968:
4901:
4881:
4731:
4646:
4574:
4501:
4409:
4299:
4250:
4164:
4119:
4084:
4037:
4017:
3985:
3965:
3945:
3705:
3679:
3641:
3601:
3569:
3509:
3412:
3369:
3334:
3302:
3158:
3114:
3041:
3021:
2967:
2940:
2894:
2826:
2766:
2720:
2626:
2578:
2558:
2502:
2480:
2451:
2399:
2368:
2348:
2284:
2240:
2207:
2179:
2054:
1947:
1920:
1888:
1742:
1643:
1512:
1435:
1403:
1377:
1308:
1256:
1204:
869:
848:
822:
703:vector addition
656:
634:
609:
284:
231:
210:
184:
159:Introductory example
149:law of large numbers
17669:Convex optimization
17433:Jensen's inequality
17303:Lagrange multiplier
17293:Convex optimization
17288:Convex metric space
17180:Business portal
17117:Operations research
16944:Substitution effect
16259:Starr, Ross M.
16195:Starr, Ross M.
16001:Varian, Hal R.
15807:Starr, Ross M.
15755:Shapley, L. S.
15596:Geometriae Dedicata
15587:Ruzsa, Imre Z.
15464:1985MPCPS..97..151P
15039:Hildenbrand, Werner
14032:Hahn, Frank H.
13966:, pp. 172–183)
13904:, pp. 478–479)
13852:, pp. 154–155)
13825:, pp. 433–434)
13801:, pp. 287–288)
13660:, pp. 458–476)
13549:, pp. 169–182)
13536:, pp. 627–630)
13531:, pp. 393–394)
13468:, pp. 217–218)
13294:, pp. 552–553)
13115:Starr, Ross M.
13034:, pp. 357–359)
13010:, pp. 435–436)
12995:, pp. 392–395)
12983:, pp. 129–130)
12843:, p. 385) and
12765:, pp. 881–882)
12742:integer constraints
12519:is a member of the
12517:limit of a sequence
12253:probability measure
12224:widely used in the
12143:convex minimization
12071:limit of a sequence
12067:convexified problem
11975:linear optimization
11869: (0, π).
11847:absolute value
11681:of a function
11598:applied mathematics
11465:, who proved their
11212:. The basket (
11092:economic equilibria
11053:, and we are done.
10156:{\displaystyle I,J}
9659:{\displaystyle E=x}
8750:
8634:
8553:
8465:
8433:{\displaystyle N+D}
8403:
8341:
8282:
8195:
8065:
7958:
7683:
7428:{\displaystyle N+D}
7392:
7331:
6296:iterative algorithm
6235:
5787: —
5419:such that, for any
5144: —
5042:balls containing it
3939: —
3110:, while the others
2888: —
990:. By definition, a
982:of members of
438:Minkowski sum
266:the sum is convex.
56:that describes the
17561:(cs, bcs)-complete
17532:Algebraic interior
17250:Convex combination
16758:Indifference curve
16726:Goods and services
16667:Economies of scope
16662:Economies of scale
16467:Algebraic geometry
16197:(September 2009).
16086:10.1007/BF00531823
15848:Blume, Lawrence E.
15646:(Economica, Paris)
15386:Mas-Colell, Andreu
15300:Mas-Colell, Andreu
15268:Mas-Colell, Andreu
15248:Lemaréchal, Claude
15070:Lemaréchal, Claude
13170:10.1007/bf01212924
12875:, p. 17) and
12859:, p. xi) and
12439:statistical theory
12281:Lyapunov's theorem
12244:. These proofs of
12198:random vector
12179:probability theory
12008:
11932:
11879:objective function
11863:sine function
11779:
11771:sine function
11638:
11606:integration theory
11570:Leonid Kantorovich
11439:
11428:Ross M. Starr
11321:
11269:indifference curve
11250:
11196:indifference curve
11176:probability theory
11152:Leonid Kantorovich
11088:Ross M. Starr
11076:
11043:
11017:
10967:
10939:
10901:
10844:
10798:
10764:
10695:
10668:
10657:
10628:
10590:
10560:
10534:
10458:
10407:
10380:
10353:
10326:
10297:
10153:
10127:
10116:
10087:
10049:
9998:
9972:
9928:
9912:
9860:
9737:
9717:
9676:
9656:
9621:
9601:
9581:
9535:
9477:
9403:
9388:
9269:
9250:
9209:
9136:
9113:
9076:
9018:
8998:
8978:
8945:
8899:
8850:
8803:
8776:
8732:
8731:
8705:
8685:
8675:where for at most
8665:
8616:
8615:
8600:
8564:that are nonzero.
8554:
8535:
8518:
8486:
8466:
8447:
8430:
8404:
8385:
8368:
8345:
8316:
8315:
8283:
8257:
8170:
8153:
8120:
8047:
8046:
8036:
8008:
7940:
7939:
7929:
7856:
7827:
7807:
7778:
7739:
7684:
7658:
7641:
7621:
7579:
7559:
7425:
7399:
7374:
7357:
7313:
7312:
7302:
7256:
7171:
7142:
7097:
7087:
7041:
7021:
6992:
6965:
6833:
6798:
6769:
6736:
6726:
6691:
6668:
6642:
6603:
6583:
6500:
6440:
6428:
6413:
6380:
6345:
6316:
6245:
6221:
6066:
6039:
6012:
5979:
5950:
5882:
5855:
5785:
5769:
5734:
5684:{\displaystyle B'}
5681:
5656:
5636:
5605:
5567:
5547:
5495:
5475:
5455:
5409:
5389:
5354:
5307:
5287:
5286:
5254:
5218:
5142:
5126:
5110:
5094:
5030:
4992:
4952:
4887:
4864:
4715:
4632:
4613:
4560:
4487:
4458:
4393:
4367:
4285:
4236:
4210:
4143:
4105:
4075:Euclidean distance
4063:
4023:
3991:
3971:
3951:
3937:
3916:
3897:
3848:
3827:
3685:
3665:
3627:
3587:
3555:
3492:
3395:
3352:
3330:and an element in
3320:
3288:
3140:
3100:
3027:
3007:
2953:
2926:
2886:
2866:
2812:
2752:
2706:
2604:
2564:
2542:
2486:
2466:
2437:
2383:
2354:
2332:
2258:
2226:
2193:
2162:
2034:
1926:
1906:
1866:
1725:
1626:
1495:
1421:
1383:
1371:
1359:
1294:
1242:
1198:Minkowski addition
1189:Minkowski addition
1122:
1118:convex combination
992:convex combination
980:convex combination
934: , the point
875:
854:
828:
764:, a point in some
672:real numbers
662:
640:
615:
467:Real vector spaces
401:
270:Minkowski addition
237:
216:
190:
141:convex preferences
137:probability theory
78:Ross M. Starr
43:
27:Minkowski addition
17639:
17638:
17198:
17197:
17160:Political economy
16959:Supply and demand
16839:Pareto efficiency
16531:
16530:
16018:978-0-393-95735-8
15936:978-0-521-08288-4
15911:Starr, R. M.
15871:978-0-333-78676-5
15738:978-3-540-78858-4
15434:978-1-84996-949-9
15399:978-0-19-507340-9
15149:, Yale University
15138:(November 1979).
15104:Hotelling, Harold
15052:978-0-691-04189-6
14872:978-0-521-31988-1
14799:978-0-444-86127-6
14611:Cassels, J. W. S.
14601:). Archived from
14214:978-0-486-46265-3
14099:Republished in a
13709:Rockafellar (1997
13633:, pp. 24–25)
13560:Hildenbrand (1974
13518:Mas-Colell (1987)
13484:, pp. 27–33)
13377:Rothenberg (
13224:, pp. 79–81)
13212:, pp. 76–79)
13100:978-1-886529-31-1
12915:, pp. 35–36)
12873:Rockafellar (1997
12861:Rockafellar (1997
12847:, pp. 11–12)
12845:Rockafellar (1997
12825:Rockafellar (1997
12823:, p. 376),
12809:Rockafellar (1997
12719:on page 381:
12547:Rockafellar (1997
12496:literary epigraph
12447:discrete analogue
12139:Claude Lemaréchal
12011:{\displaystyle N}
11994:, ...,
11935:{\displaystyle N}
11829:For example, the
11650:iterative methods
11588:), as well as in
11550:Tjalling Koopmans
11132:Tjalling Koopmans
11005:
10889:
10829:
10783:
10749:
10642:
10613:
10101:
10072:
9960:
9926:
9740:{\displaystyle o}
9727:centered at some
9679:{\displaystyle S}
9624:{\displaystyle S}
9604:{\displaystyle X}
9401:
9244:
9210:
9174:
9139:{\displaystyle S}
9021:{\displaystyle S}
9001:{\displaystyle X}
8820:We can interpret
8761:
8722:
8708:{\displaystyle n}
8688:{\displaystyle D}
8645:
8606:
8591:
8585:
8489:{\displaystyle D}
8371:{\displaystyle n}
8306:
8209:
8079:
8037:
8027:
7969:
7930:
7920:
7885:
7853:
7830:{\displaystyle 1}
7804:
7713:
7644:{\displaystyle k}
7582:{\displaystyle x}
7506:
7453:
7342:
7303:
7293:
7287:
7201:
7168:
7124:
7088:
7078:
7072:
7044:{\displaystyle n}
7002:is the vector in
6912:
6859:
6727:
6717:
6659:
6606:{\displaystyle K}
6443:{\displaystyle n}
6426:
6319:{\displaystyle x}
6240:
6174:
6125:
5843:
5783:
5659:{\displaystyle B}
5646:is the radius of
5569:For example, let
5498:{\displaystyle r}
5478:{\displaystyle B}
5412:{\displaystyle r}
5310:{\displaystyle D}
5274:
5206:
5205:
5199:
5140:
5136:Now we can state
5095:
5072:
4890:{\displaystyle Q}
4598:
4443:
4335:
4195:
4026:{\displaystyle Q}
3994:{\displaystyle D}
3974:{\displaystyle D}
3954:{\displaystyle D}
3935:
3882:
3833:
3806:
3766:
3688:{\displaystyle x}
3030:{\displaystyle D}
2884:
2567:{\displaystyle x}
2489:{\displaystyle N}
2357:{\displaystyle D}
1929:{\displaystyle X}
1844:
1479:
1386:{\displaystyle X}
1149:. Thus Conv(
799:is defined to be
665:{\displaystyle D}
643:{\displaystyle D}
618:{\displaystyle D}
17701:
17557:(cs, lcs)-closed
17503:Effective domain
17458:Robinson–Ursescu
17334:Convex conjugate
17225:
17218:
17211:
17202:
17201:
17188:
17187:
17178:
17177:
16920:Returns to scale
16778:Market structure
16558:
16551:
16544:
16535:
16534:
16521:
16520:
16309:
16302:
16295:
16286:
16285:
16281:
16279:
16277:
16267:
16254:
16252:
16250:
16244:
16212:. pp. 3–6.
16207:
16190:
16188:
16186:
16176:
16158:
16138:
16098:
16088:
16059:
16030:
16010:
15996:
15969:
15940:
15906:
15875:
15834:
15802:
15797:. Archived from
15761:(October 1966).
15750:
15715:
15686:
15641:
15622:
15612:
15582:
15553:
15513:
15483:
15446:
15414:
15403:
15381:
15329:
15295:
15263:
15243:
15211:
15185:
15158:
15156:
15154:
15144:
15131:
15099:
15064:
15034:
15009:Guesnerie, Roger
15004:
14965:
14936:
14907:
14876:
14849:
14811:
14768:
14729:
14705:
14695:
14686:(8): 2431–2436.
14668:
14638:
14606:
14595:Author's website
14565:
14532:
14530:
14528:
14507:
14493:
14463:
14441:
14412:
14383:
14343:
14333:
14305:
14265:
14226:
14199:
14165:
14138:
14133:. Archived from
14098:
14061:
14023:
14021:
14019:
14005:
13979:
13976:Mas-Colell (1978
13973:
13967:
13961:
13955:
13911:
13905:
13899:
13893:
13887:
13881:
13871:
13865:
13859:
13853:
13847:
13838:
13832:
13826:
13820:
13814:
13808:
13802:
13796:
13790:
13784:
13778:
13777:
13745:
13736:
13721:Lemaréchal (1973
13718:
13712:
13706:
13700:
13694:
13685:
13679:
13673:
13667:
13661:
13655:
13646:
13640:
13634:
13628:
13622:
13616:
13610:
13609:
13582:
13576:
13554:Mas-Colell (1985
13544:
13538:
13526:
13520:
13515:
13509:
13503:
13497:
13491:
13485:
13475:
13469:
13463:
13454:
13439:
13433:
13419:
13410:
13404:
13398:
13392:
13386:
13375:
13369:
13358:Rothenberg (1960
13352:, p. 484),
13339:
13333:
13322:
13316:
13301:
13295:
13289:
13283:
13271:
13265:
13255:
13249:
13243:
13237:
13231:
13225:
13219:
13213:
13206:Mas-Colell (1985
13203:
13197:
13191:
13182:
13181:
13149:
13143:
13142:
13111:
13105:
13104:
13083:
13077:
13076:
13074:
13072:
13062:
13053:
13047:
13041:
13035:
13029:
13023:
13017:
13011:
13005:
12996:
12990:
12984:
12978:
12972:
12966:
12955:
12949:
12943:
12937:
12931:
12922:
12916:
12910:
12904:
12898:
12892:
12886:
12880:
12870:
12864:
12854:
12848:
12838:
12832:
12818:
12812:
12806:
12797:
12791:
12785:
12779:
12766:
12760:
12751:
12706:
12697:
12687:
12672:
12667:
12656:
12650:
12641:
12636:
12617:
12556:
12550:
12513:
12507:
12462:
12356:
12340:
12323:product function
12273:Lebesgue measure
12238:large-deviations
12222:
12039:
12017:
12015:
12014:
12009:
11941:
11939:
11938:
11933:
11861:|. However, the
11582:public economics
11540:: Arrow (1972),
11474:quasi-equilibria
11455:aggregate demand
11351:Harold Hotelling
11052:
11050:
11049:
11044:
11039:
11038:
11016:
10976:
10974:
10973:
10968:
10948:
10946:
10945:
10940:
10923:
10922:
10900:
10885:
10881:
10871:
10870:
10843:
10825:
10821:
10820:
10797:
10779:
10775:
10774:
10773:
10763:
10704:
10702:
10701:
10696:
10694:
10693:
10677:
10675:
10674:
10669:
10667:
10666:
10656:
10638:
10637:
10627:
10599:
10597:
10596:
10591:
10589:
10588:
10569:
10567:
10566:
10561:
10543:
10541:
10540:
10535:
10524:
10523:
10496:
10495:
10467:
10465:
10464:
10459:
10457:
10456:
10441:
10440:
10428:
10416:
10414:
10413:
10408:
10406:
10405:
10389:
10387:
10386:
10381:
10379:
10378:
10362:
10360:
10359:
10354:
10352:
10351:
10335:
10333:
10332:
10327:
10306:
10304:
10303:
10298:
10296:
10295:
10283:
10282:
10267:
10266:
10254:
10237:
10236:
10218:
10210:
10162:
10160:
10159:
10154:
10136:
10134:
10133:
10128:
10126:
10125:
10115:
10097:
10096:
10086:
10058:
10056:
10055:
10050:
10039:
10007:
10005:
10004:
9999:
9994:
9993:
9971:
9956:
9921:
9919:
9918:
9913:
9869:
9867:
9866:
9861:
9829:
9828:
9746:
9744:
9743:
9738:
9726:
9724:
9723:
9718:
9685:
9683:
9682:
9677:
9665:
9663:
9662:
9657:
9630:
9628:
9627:
9622:
9610:
9608:
9607:
9602:
9590:
9588:
9587:
9582:
9571:
9544:
9542:
9541:
9536:
9486:
9484:
9483:
9478:
9455:
9454:
9436:
9397:
9395:
9394:
9389:
9330:
9329:
9311:
9278:
9276:
9275:
9270:
9249:
9245:
9242:
9219:
9208:
9198:
9145:
9143:
9142:
9137:
9122:
9120:
9119:
9114:
9103:
9085:
9083:
9082:
9077:
9075:
9074:
9059:
9058:
9027:
9025:
9024:
9019:
9007:
9005:
9004:
8999:
8987:
8985:
8984:
8979:
8971:
8970:
8954:
8952:
8951:
8946:
8944:
8943:
8934:
8933:
8908:
8906:
8905:
8900:
8889:
8859:
8857:
8856:
8851:
8840:
8812:
8810:
8809:
8804:
8802:
8801:
8785:
8783:
8782:
8777:
8775:
8774:
8763:
8762:
8754:
8746:
8730:
8714:
8712:
8711:
8706:
8694:
8692:
8691:
8686:
8674:
8672:
8671:
8666:
8664:
8660:
8659:
8658:
8647:
8646:
8638:
8630:
8614:
8599:
8587:
8586:
8578:
8563:
8561:
8560:
8555:
8549:
8527:
8525:
8524:
8519:
8495:
8493:
8492:
8487:
8475:
8473:
8472:
8467:
8461:
8439:
8437:
8436:
8431:
8413:
8411:
8410:
8405:
8399:
8377:
8375:
8374:
8369:
8354:
8352:
8351:
8346:
8337:
8330:
8329:
8314:
8292:
8290:
8289:
8284:
8278:
8271:
8270:
8253:
8252:
8251:
8250:
8230:
8229:
8222:
8221:
8211:
8210:
8202:
8191:
8184:
8183:
8162:
8160:
8159:
8154:
8152:
8151:
8129:
8127:
8126:
8121:
8116:
8115:
8114:
8113:
8093:
8092:
8081:
8080:
8072:
8061:
8045:
8035:
8017:
8015:
8014:
8009:
8007:
8006:
8005:
8004:
7988:
7984:
7983:
7982:
7971:
7970:
7962:
7954:
7938:
7928:
7910:
7909:
7908:
7907:
7887:
7886:
7878:
7865:
7863:
7862:
7857:
7855:
7854:
7846:
7836:
7834:
7833:
7828:
7816:
7814:
7813:
7808:
7806:
7805:
7797:
7787:
7785:
7784:
7779:
7774:
7773:
7748:
7746:
7745:
7740:
7738:
7737:
7736:
7735:
7715:
7714:
7706:
7693:
7691:
7690:
7685:
7679:
7672:
7671:
7650:
7648:
7647:
7642:
7630:
7628:
7627:
7622:
7608:
7607:
7588:
7586:
7585:
7580:
7568:
7566:
7565:
7560:
7555:
7554:
7542:
7541:
7520:
7519:
7508:
7507:
7499:
7455:
7454:
7446:
7434:
7432:
7431:
7426:
7408:
7406:
7405:
7400:
7388:
7366:
7364:
7363:
7358:
7356:
7355:
7344:
7343:
7335:
7327:
7311:
7301:
7289:
7288:
7280:
7265:
7263:
7262:
7257:
7255:
7254:
7215:
7214:
7203:
7202:
7194:
7180:
7178:
7177:
7172:
7170:
7169:
7161:
7151:
7149:
7148:
7143:
7138:
7137:
7126:
7125:
7117:
7113:
7112:
7096:
7086:
7074:
7073:
7065:
7050:
7048:
7047:
7042:
7030:
7028:
7027:
7022:
7020:
7019:
7014:
7001:
6999:
6998:
6993:
6991:
6990:
6974:
6972:
6971:
6966:
6961:
6960:
6948:
6947:
6926:
6925:
6914:
6913:
6905:
6861:
6860:
6852:
6842:
6840:
6839:
6834:
6832:
6831:
6820:
6807:
6805:
6804:
6799:
6797:
6796:
6791:
6778:
6776:
6775:
6770:
6768:
6767:
6752:
6751:
6735:
6725:
6700:
6698:
6697:
6692:
6684:
6683:
6667:
6651:
6649:
6648:
6643:
6635:
6634:
6612:
6610:
6609:
6604:
6592:
6590:
6589:
6584:
6582:
6581:
6566:
6565:
6549:
6544:
6526:
6525:
6509:
6507:
6506:
6501:
6496:
6495:
6483:
6466:
6465:
6449:
6447:
6446:
6441:
6422:
6420:
6419:
6414:
6412:
6411:
6406:
6389:
6387:
6386:
6381:
6379:
6378:
6367:
6354:
6352:
6351:
6346:
6344:
6343:
6338:
6325:
6323:
6322:
6317:
6254:
6252:
6251:
6246:
6241:
6236:
6234:
6229:
6216:
6211:
6207:
6206:
6205:
6195:
6190:
6175:
6167:
6162:
6158:
6157:
6156:
6146:
6141:
6126:
6118:
6111:
6092:
6091:
6075:
6073:
6072:
6067:
6065:
6064:
6048:
6046:
6045:
6040:
6038:
6037:
6021:
6019:
6018:
6013:
6005:
6004:
5988:
5986:
5985:
5980:
5978:
5977:
5972:
5959:
5957:
5956:
5951:
5949:
5948:
5915:
5914:
5891:
5889:
5888:
5883:
5881:
5880:
5871:
5870:
5854:
5827:
5804:
5803:
5788:
5778:
5776:
5775:
5770:
5768:
5767:
5762:
5743:
5741:
5740:
5735:
5690:
5688:
5687:
5682:
5680:
5665:
5663:
5662:
5657:
5645:
5643:
5642:
5637:
5635:
5614:
5612:
5611:
5606:
5604:
5603:
5598:
5583:
5556:
5554:
5553:
5548:
5531:
5504:
5502:
5501:
5496:
5484:
5482:
5481:
5476:
5464:
5462:
5461:
5456:
5445:
5418:
5416:
5415:
5410:
5398:
5396:
5395:
5390:
5388:
5387:
5382:
5363:
5361:
5360:
5355:
5317:largest terms".
5316:
5314:
5313:
5308:
5296:
5294:
5293:
5288:
5285:
5263:
5261:
5260:
5255:
5250:
5249:
5240:
5239:
5217:
5203:
5197:
5184:
5161:
5160:
5145:
5135:
5133:
5132:
5127:
5109:
5093:
5092:
5091:
5039:
5037:
5036:
5031:
5001:
4999:
4998:
4993:
4988:
4987:
4982:
4961:
4959:
4958:
4953:
4930:
4913:
4912:
4896:
4894:
4893:
4888:
4873:
4871:
4870:
4865:
4860:
4859:
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4742:
4724:
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4721:
4716:
4711:
4710:
4701:
4675:
4658:
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4569:
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4496:
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4457:
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4388:
4366:
4365:
4328:
4311:
4310:
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4280:
4275:
4266:
4245:
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4232:
4231:
4209:
4176:
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4152:
4150:
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4144:
4139:
4138:
4133:
4114:
4112:
4111:
4106:
4104:
4103:
4098:
4072:
4070:
4069:
4064:
4053:
4032:
4030:
4029:
4024:
4000:
3998:
3997:
3992:
3980:
3978:
3977:
3972:
3960:
3958:
3957:
3952:
3940:
3925:
3923:
3922:
3917:
3912:
3908:
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3896:
3875:
3874:
3862:
3847:
3826:
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3811:
3804:
3762:
3758:
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3756:
3746:
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3721:
3694:
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3691:
3686:
3674:
3672:
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3666:
3636:
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3633:
3628:
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3625:
3613:
3612:
3596:
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3593:
3588:
3564:
3562:
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3556:
3551:
3550:
3538:
3521:
3520:
3501:
3499:
3498:
3493:
3491:
3490:
3480:
3475:
3451:
3450:
3440:
3435:
3404:
3402:
3401:
3396:
3385:
3361:
3359:
3358:
3355:{\displaystyle }
3353:
3329:
3327:
3326:
3321:
3297:
3295:
3294:
3289:
3266:
3228:
3149:
3147:
3146:
3141:
3139:
3138:
3126:
3125:
3109:
3107:
3106:
3101:
3099:
3098:
3083:
3082:
3070:
3053:
3052:
3037:of the summands
3036:
3034:
3033:
3028:
3016:
3014:
3013:
3008:
3000:
2999:
2989:
2984:
2962:
2960:
2959:
2954:
2952:
2951:
2935:
2933:
2932:
2927:
2916:
2889:
2875:
2873:
2872:
2867:
2859:
2858:
2848:
2843:
2821:
2819:
2818:
2813:
2808:
2807:
2795:
2778:
2777:
2761:
2759:
2758:
2753:
2742:
2715:
2713:
2712:
2707:
2702:
2701:
2689:
2674:
2669:
2642:
2613:
2611:
2610:
2605:
2594:
2573:
2571:
2570:
2565:
2551:
2549:
2548:
2543:
2541:
2540:
2530:
2525:
2495:
2493:
2492:
2487:
2475:
2473:
2472:
2467:
2465:
2464:
2459:
2446:
2444:
2443:
2438:
2436:
2435:
2411:
2410:
2392:
2390:
2389:
2384:
2382:
2381:
2376:
2363:
2361:
2360:
2355:
2341:
2339:
2338:
2333:
2267:
2265:
2264:
2259:
2235:
2233:
2232:
2227:
2219:
2218:
2202:
2200:
2199:
2194:
2192:
2171:
2169:
2168:
2163:
2158:
2157:
2145:
2130:
2125:
2104:
2103:
2093:
2088:
2070:
2043:
2041:
2040:
2035:
2021:
1995:
1963:
1935:
1933:
1932:
1927:
1915:
1913:
1912:
1907:
1875:
1873:
1872:
1867:
1842:
1838:
1837:
1825:
1824:
1812:
1811:
1801:
1796:
1775:
1774:
1764:
1759:
1734:
1732:
1731:
1726:
1721:
1720:
1702:
1701:
1689:
1688:
1676:
1675:
1665:
1660:
1635:
1633:
1632:
1627:
1504:
1502:
1501:
1496:
1477:
1430:
1428:
1427:
1422:
1392:
1390:
1389:
1384:
1368:
1366:
1365:
1360:
1358:
1357:
1333:
1332:
1320:
1319:
1303:
1301:
1300:
1295:
1293:
1292:
1268:
1267:
1251:
1249:
1248:
1243:
1241:
1240:
1216:
1215:
1140:
1087: = 1.
1086:
1056:
884:
882:
881:
876:
863:
861:
860:
855:
839:is convex but a
837:
835:
834:
829:
753:
725:
696:
671:
669:
668:
663:
649:
647:
646:
641:
624:
622:
621:
616:
578:coordinate-wise
504:coordinate-wise
410:
408:
407:
402:
246:
244:
243:
238:
225:
223:
222:
217:
199:
197:
196:
191:
117:(alternatively,
96:Shapley–Folkman
17709:
17708:
17704:
17703:
17702:
17700:
17699:
17698:
17664:Convex geometry
17644:
17643:
17640:
17635:
17619:
17586:
17541:
17472:
17398:
17389:Semi-continuity
17374:Convex function
17355:Logarithmically
17322:
17283:Convex geometry
17264:
17255:Convex function
17238:
17232:Convex analysis
17229:
17199:
17194:
17172:
17164:
17131:
17010:
16652:Deadweight loss
16589:Consumer choice
16567:
16562:
16532:
16527:
16509:
16476:
16445:
16402:
16395:
16350:
16343:
16318:
16313:
16275:
16273:
16265:
16248:
16246:
16242:
16228:
16205:
16184:
16182:
16174:
16166:
16161:
16048:10.2307/2525560
16019:
15985:
15958:10.1137/0328026
15937:
15895:
15872:
15823:10.2307/1909201
15779:10.2307/1910101
15739:
15704:
15667:10.2307/2549499
15661:(68): 355–385.
15638:
15502:
15492:Convex analysis
15435:
15400:
15374:
15346:Milgate, Murray
15338:"Non-convexity"
15318:
15256:Le Chesnay
15208:
15174:
15152:
15150:
15142:
15120:10.2307/1907346
15088:
15053:
15023:
14993:
14873:
14838:
14800:
14657:
14583:
14526:
14524:
14505:
14482:
14460:
14364:10.2307/1909854
14312:(August 1965).
14286:10.2307/1913732
14215:
14137:on 24 May 2011.
14131:
14114:
14079:10.1137/1022026
14050:
14017:
14015:
14003:
13987:
13982:
13974:
13970:
13962:
13958:
13912:
13908:
13900:
13896:
13888:
13884:
13872:
13868:
13860:
13856:
13848:
13841:
13835:Molchanov (2005
13833:
13829:
13821:
13817:
13809:
13805:
13797:
13793:
13785:
13781:
13746:
13739:
13728:, pp. 2–3)
13719:
13715:
13707:
13703:
13697:Bertsekas (1999
13695:
13688:
13680:
13676:
13668:
13664:
13656:
13649:
13641:
13637:
13629:
13625:
13617:
13613:
13606:
13583:
13579:
13572:Ellickson (1994
13545:
13541:
13527:
13523:
13516:
13512:
13506:Guesnerie (1989
13504:
13500:
13492:
13488:
13476:
13472:
13464:
13457:
13453:, p. 552).
13440:
13436:
13420:
13413:
13405:
13401:
13393:
13389:
13376:
13372:
13340:
13336:
13323:
13319:
13302:
13298:
13290:
13286:
13274:Samuelson (1950
13272:
13268:
13256:
13252:
13246:Hotelling (1935
13244:
13240:
13232:
13228:
13220:
13216:
13204:
13200:
13194:Guesnerie (1989
13192:
13185:
13158:Economic Theory
13150:
13146:
13112:
13108:
13101:
13084:
13080:
13070:
13068:
13060:
13054:
13050:
13042:
13038:
13030:
13026:
13020:Schneider (1993
13018:
13014:
13006:
12999:
12991:
12987:
12981:Schneider (1993
12979:
12975:
12967:
12958:
12952:Schneider (1993
12950:
12946:
12938:
12934:
12925:Schneider (1993
12923:
12919:
12911:
12907:
12899:
12895:
12891:, pp. 2–3)
12889:Schneider (1993
12887:
12883:
12871:
12867:
12857:Schneider (1993
12855:
12851:
12839:
12835:
12819:
12815:
12807:
12800:
12792:
12788:
12780:
12769:
12761:
12754:
12728:methods to the
12726:Lagrangian dual
12722:Bertsekas (1996
12709:Bertsekas (1996
12707:
12700:
12688:
12675:
12668:
12659:
12651:
12644:
12637:
12624:
12620:
12614:level sets
12557:
12553:
12514:
12510:
12490:
12488:
12463:
12459:
12455:
12420:
12412:
12401:
12395:
12387:
12384:
12377:
12371:
12355:
12348:
12342:
12339:
12332:
12326:
12312:
12305:
12277:Euclidean space
12265:vector measures
12220:
12175:
12125:
12121:
12112:
12108:
12105:
12102:
12098:
12087:
12079:
12064:
12060:
12049:
12043:
12037:
12027:
12022:
12018:
12003:
12000:
11999:
11993:
11986:
11968:
11959:
11950:
11946:
11942:
11927:
11924:
11923:
11917:
11910:
11875:
11857:) = |
11822:convex function
11813:
11789:
11738:
11737:
11733:
11718:
11715:
11714:
11662:
11654:convex problems
11646:non-convexities
11614:
11586:market failures
11538:Nobel laureates
11503:
11493:
11482:
11413:
11327:
11313:
11262:, a consumer's
11256:
11247:
11240:
11225:
11218:
11211:
11204:
11184:
11164:
11120:Nobel laureates
11060:
11055:
11034:
11030:
11009:
10982:
10979:
10978:
10956:
10953:
10952:
10918:
10914:
10893:
10866:
10862:
10849:
10845:
10833:
10816:
10812:
10808:
10787:
10769:
10765:
10753:
10748:
10744:
10712:
10709:
10708:
10689:
10685:
10683:
10680:
10679:
10662:
10658:
10646:
10633:
10629:
10617:
10605:
10602:
10601:
10584:
10580:
10578:
10575:
10574:
10549:
10546:
10545:
10519:
10515:
10491:
10487:
10473:
10470:
10469:
10452:
10448:
10436:
10432:
10424:
10422:
10419:
10418:
10401:
10397:
10395:
10392:
10391:
10374:
10370:
10368:
10365:
10364:
10347:
10343:
10341:
10338:
10337:
10315:
10312:
10311:
10291:
10287:
10278:
10274:
10262:
10258:
10241:
10232:
10228:
10214:
10206:
10168:
10165:
10164:
10142:
10139:
10138:
10121:
10117:
10105:
10092:
10088:
10076:
10064:
10061:
10060:
10026:
10015:
10012:
10011:
9989:
9985:
9964:
9946:
9935:
9932:
9931:
9924:
9877:
9874:
9873:
9824:
9820:
9752:
9749:
9748:
9732:
9729:
9728:
9691:
9688:
9687:
9671:
9668:
9667:
9636:
9633:
9632:
9616:
9613:
9612:
9596:
9593:
9592:
9558:
9550:
9547:
9546:
9494:
9491:
9490:
9450:
9446:
9423:
9409:
9406:
9405:
9325:
9321:
9298:
9284:
9281:
9280:
9241:
9215:
9214:
9185:
9178:
9151:
9148:
9147:
9131:
9128:
9127:
9099:
9091:
9088:
9087:
9070:
9066:
9054:
9050:
9033:
9030:
9029:
9013:
9010:
9009:
8993:
8990:
8989:
8966:
8962:
8960:
8957:
8956:
8939:
8935:
8929:
8925:
8914:
8911:
8910:
8876:
8865:
8862:
8861:
8827:
8825:
8822:
8821:
8815:
8797:
8793:
8791:
8788:
8787:
8764:
8753:
8752:
8751:
8736:
8726:
8720:
8717:
8716:
8700:
8697:
8696:
8680:
8677:
8676:
8648:
8637:
8636:
8635:
8620:
8610:
8605:
8601:
8595:
8577:
8576:
8574:
8571:
8570:
8539:
8533:
8530:
8529:
8501:
8498:
8497:
8481:
8478:
8477:
8451:
8445:
8442:
8441:
8419:
8416:
8415:
8389:
8383:
8380:
8379:
8363:
8360:
8359:
8325:
8321:
8320:
8310:
8298:
8295:
8294:
8266:
8262:
8261:
8246:
8242:
8235:
8231:
8217:
8213:
8212:
8201:
8200:
8199:
8179:
8175:
8174:
8168:
8165:
8164:
8147:
8143:
8135:
8132:
8131:
8109:
8105:
8098:
8094:
8082:
8071:
8070:
8069:
8051:
8041:
8031:
8022:
8019:
8018:
8000:
7996:
7989:
7972:
7961:
7960:
7959:
7944:
7934:
7924:
7919:
7915:
7914:
7903:
7899:
7892:
7888:
7877:
7876:
7871:
7868:
7867:
7845:
7844:
7842:
7839:
7838:
7822:
7819:
7818:
7796:
7795:
7793:
7790:
7789:
7769:
7765:
7754:
7751:
7750:
7731:
7727:
7720:
7716:
7705:
7704:
7699:
7696:
7695:
7667:
7663:
7662:
7656:
7653:
7652:
7636:
7633:
7632:
7603:
7599:
7597:
7594:
7593:
7574:
7571:
7570:
7550:
7546:
7531:
7527:
7509:
7498:
7497:
7496:
7445:
7444:
7442:
7439:
7438:
7414:
7411:
7410:
7378:
7372:
7369:
7368:
7345:
7334:
7333:
7332:
7317:
7307:
7297:
7279:
7278:
7276:
7273:
7272:
7220:
7216:
7204:
7193:
7192:
7191:
7186:
7183:
7182:
7160:
7159:
7157:
7154:
7153:
7127:
7116:
7115:
7114:
7102:
7098:
7092:
7082:
7064:
7063:
7061:
7058:
7057:
7036:
7033:
7032:
7015:
7010:
7009:
7007:
7004:
7003:
6986:
6982:
6980:
6977:
6976:
6956:
6952:
6937:
6933:
6915:
6904:
6903:
6902:
6851:
6850:
6848:
6845:
6844:
6821:
6816:
6815:
6813:
6810:
6809:
6792:
6787:
6786:
6784:
6781:
6780:
6757:
6753:
6741:
6737:
6731:
6721:
6709:
6706:
6705:
6673:
6669:
6663:
6657:
6654:
6653:
6624:
6620:
6618:
6615:
6614:
6598:
6595:
6594:
6571:
6567:
6555:
6551:
6545:
6534:
6521:
6517:
6515:
6512:
6511:
6491:
6487:
6470:
6461:
6457:
6455:
6452:
6451:
6435:
6432:
6431:
6407:
6402:
6401:
6399:
6396:
6395:
6368:
6363:
6362:
6360:
6357:
6356:
6339:
6334:
6333:
6331:
6328:
6327:
6311:
6308:
6307:
6304:
6261:
6230:
6225:
6217:
6215:
6201:
6197:
6191:
6180:
6166:
6152:
6148:
6142:
6131:
6117:
6116:
6112:
6098:
6097:
6093:
6087:
6083:
6081:
6078:
6077:
6060:
6056:
6054:
6051:
6050:
6033:
6029:
6027:
6024:
6023:
6000:
5996:
5994:
5991:
5990:
5973:
5968:
5967:
5965:
5962:
5961:
5920:
5916:
5910:
5906:
5901:
5898:
5897:
5894:
5876:
5872:
5866:
5862:
5847:
5814:
5799:
5795:
5793:
5790:
5789:
5786:
5763:
5758:
5757:
5749:
5746:
5745:
5699:
5696:
5695:
5673:
5671:
5668:
5667:
5651:
5648:
5647:
5628:
5620:
5617:
5616:
5599:
5594:
5593:
5576:
5574:
5571:
5570:
5518:
5510:
5507:
5506:
5490:
5487:
5486:
5470:
5467:
5466:
5432:
5424:
5421:
5420:
5404:
5401:
5400:
5383:
5378:
5377:
5369:
5366:
5365:
5340:
5337:
5336:
5330:
5302:
5299:
5298:
5278:
5272:
5269:
5268:
5265:
5245:
5241:
5235:
5231:
5210:
5171:
5156:
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5147:
5146:
5143:
5099:
5087:
5083:
5076:
5049:
5046:
5045:
5010:
5007:
5006:
4983:
4978:
4977:
4969:
4966:
4965:
4917:
4908:
4904:
4902:
4899:
4898:
4882:
4879:
4878:
4855:
4851:
4795:
4738:
4734:
4732:
4729:
4728:
4706:
4702:
4694:
4668:
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4649:
4647:
4644:
4643:
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4575:
4572:
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4551:
4547:
4508:
4504:
4502:
4499:
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4410:
4407:
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4384:
4380:
4358:
4339:
4321:
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4302:
4300:
4297:
4296:
4276:
4271:
4270:
4259:
4251:
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4247:
4227:
4223:
4199:
4171:
4167:
4165:
4162:
4161:
4134:
4129:
4128:
4120:
4117:
4116:
4099:
4094:
4093:
4085:
4082:
4081:
4040:
4038:
4035:
4034:
4018:
4015:
4014:
4011:
4003:
3986:
3983:
3982:
3966:
3963:
3962:
3946:
3943:
3942:
3938:
3932:
3902:
3898:
3886:
3870:
3866:
3849:
3837:
3832:
3828:
3815:
3807:
3770:
3752:
3748:
3742:
3731:
3726:
3722:
3708:
3706:
3703:
3702:
3680:
3677:
3676:
3642:
3639:
3638:
3621:
3617:
3608:
3604:
3602:
3599:
3598:
3570:
3567:
3566:
3546:
3542:
3525:
3516:
3512:
3510:
3507:
3506:
3486:
3482:
3476:
3459:
3446:
3442:
3436:
3425:
3413:
3410:
3409:
3372:
3370:
3367:
3366:
3335:
3332:
3331:
3303:
3300:
3299:
3253:
3215:
3159:
3156:
3155:
3152:
3134:
3130:
3121:
3117:
3115:
3112:
3111:
3094:
3090:
3078:
3074:
3057:
3048:
3044:
3042:
3039:
3038:
3022:
3019:
3018:
2995:
2991:
2985:
2974:
2968:
2965:
2964:
2947:
2943:
2941:
2938:
2937:
2903:
2895:
2892:
2891:
2887:
2854:
2850:
2844:
2833:
2827:
2824:
2823:
2803:
2799:
2782:
2773:
2769:
2767:
2764:
2763:
2729:
2721:
2718:
2717:
2697:
2693:
2676:
2670:
2659:
2629:
2627:
2624:
2623:
2620:
2581:
2579:
2576:
2575:
2559:
2556:
2555:
2536:
2532:
2526:
2515:
2503:
2500:
2499:
2481:
2478:
2477:
2460:
2455:
2454:
2452:
2449:
2448:
2431:
2427:
2406:
2402:
2400:
2397:
2396:
2377:
2372:
2371:
2369:
2366:
2365:
2349:
2346:
2345:
2285:
2282:
2281:
2279:
2274:
2241:
2238:
2237:
2214:
2210:
2208:
2205:
2204:
2188:
2180:
2177:
2176:
2153:
2149:
2132:
2126:
2115:
2099:
2095:
2089:
2078:
2057:
2055:
2052:
2051:
2008:
1982:
1950:
1948:
1945:
1944:
1921:
1918:
1917:
1889:
1886:
1885:
1882:
1833:
1829:
1820:
1816:
1807:
1803:
1797:
1786:
1770:
1766:
1760:
1749:
1743:
1740:
1739:
1716:
1712:
1697:
1693:
1684:
1680:
1671:
1667:
1661:
1650:
1644:
1641:
1640:
1513:
1510:
1509:
1436:
1433:
1432:
1404:
1401:
1400:
1378:
1375:
1374:
1353:
1349:
1328:
1324:
1315:
1311:
1309:
1306:
1305:
1288:
1284:
1263:
1259:
1257:
1254:
1253:
1236:
1232:
1211:
1207:
1205:
1202:
1201:
1191:
1129:
1106:
1085:
1078:
1071:
1065:
1063:
1054:
1048:
1041:
1035:
1028:
1022:
1016:
1014:
1007:
1000:
955:
948:
938:(1 −
921:
914:
870:
867:
866:
864:
849:
846:
845:
838:
823:
820:
819:
789:
788:
787:
786:
775:
774:
773:
754:
746:
745:
726:
715:
694:
687:
680:
673:
657:
654:
653:
635:
632:
631:
610:
607:
606:
563:
556:
549:
542:
535:
528:
521:
514:
482:can be given a
469:
461:convex geometry
457:
285:
282:
281:
232:
229:
228:
226:
211:
208:
207:
200:
185:
182:
181:
161:
54:convex geometry
52:is a result in
35:non-convex sets
17:
12:
11:
5:
17707:
17697:
17696:
17691:
17686:
17681:
17676:
17671:
17666:
17661:
17656:
17637:
17636:
17634:
17633:
17627:
17625:
17621:
17620:
17618:
17617:
17612:
17610:Strong duality
17607:
17602:
17596:
17594:
17588:
17587:
17585:
17584:
17549:
17547:
17543:
17542:
17540:
17539:
17534:
17525:
17520:
17518:John ellipsoid
17515:
17510:
17505:
17500:
17486:
17480:
17478:
17474:
17473:
17471:
17470:
17465:
17460:
17455:
17450:
17445:
17440:
17435:
17430:
17425:
17420:
17415:
17409:
17407:
17405:results (list)
17400:
17399:
17397:
17396:
17391:
17386:
17381:
17379:Invex function
17376:
17367:
17362:
17357:
17352:
17347:
17341:
17336:
17330:
17328:
17324:
17323:
17321:
17320:
17315:
17310:
17305:
17300:
17295:
17290:
17285:
17280:
17278:Choquet theory
17274:
17272:
17266:
17265:
17263:
17262:
17257:
17252:
17246:
17244:
17243:Basic concepts
17240:
17239:
17228:
17227:
17220:
17213:
17205:
17196:
17195:
17193:
17192:
17182:
17169:
17166:
17165:
17163:
17162:
17157:
17155:Macroeconomics
17152:
17151:
17150:
17139:
17137:
17133:
17132:
17130:
17129:
17124:
17119:
17114:
17109:
17104:
17099:
17094:
17089:
17084:
17079:
17074:
17069:
17064:
17059:
17054:
17049:
17044:
17039:
17034:
17029:
17024:
17018:
17016:
17012:
17011:
17009:
17008:
17003:
17002:
17001:
16996:
16986:
16981:
16980:
16979:
16970:
16956:
16951:
16946:
16941:
16932:
16927:
16922:
16917:
16912:
16907:
16902:
16897:
16892:
16891:
16890:
16885:
16876:
16871:
16866:
16861:
16856:
16854:Price controls
16846:
16841:
16836:
16835:
16834:
16829:
16824:
16819:
16818:
16817:
16812:
16802:
16797:
16796:
16795:
16790:
16775:
16773:Market failure
16770:
16765:
16760:
16755:
16750:
16745:
16740:
16739:
16738:
16733:
16723:
16718:
16713:
16708:
16707:
16706:
16696:
16695:
16694:
16689:
16684:
16679:
16669:
16664:
16659:
16654:
16649:
16644:
16643:
16642:
16637:
16632:
16627:
16626:
16625:
16615:
16610:
16600:
16591:
16586:
16581:
16575:
16573:
16569:
16568:
16565:Microeconomics
16561:
16560:
16553:
16546:
16538:
16529:
16528:
16526:
16525:
16514:
16511:
16510:
16508:
16507:
16502:
16497:
16496:
16495:
16484:
16482:
16478:
16477:
16475:
16474:
16469:
16464:
16459:
16453:
16451:
16447:
16446:
16444:
16443:
16438:
16433:
16428:
16423:
16418:
16413:
16407:
16405:
16401:Non-Euclidean
16397:
16396:
16394:
16393:
16391:Solid geometry
16388:
16387:
16386:
16381:
16374:Plane geometry
16371:
16366:
16361:
16355:
16353:
16345:
16344:
16342:
16341:
16336:
16335:
16334:
16323:
16320:
16319:
16312:
16311:
16304:
16297:
16289:
16283:
16282:
16255:
16245:on 1 July 2010
16226:
16191:
16165:
16164:External links
16162:
16160:
16159:
16139:
16099:
16079:(2): 203–208.
16060:
16042:(2): 165–177.
16031:
16017:
15997:
15983:
15970:
15952:(2): 478–481.
15941:
15935:
15907:
15893:
15876:
15870:
15835:
15803:
15773:(4): 805–827.
15751:
15737:
15716:
15702:
15687:
15657:. New Series.
15647:
15636:
15623:
15603:(3): 337–348.
15583:
15571:10.1086/258543
15565:(5): 490–492.
15554:
15542:10.1086/258363
15536:(5): 435–468.
15525:
15500:
15484:
15458:(1): 151–158.
15447:
15433:
15404:
15398:
15382:
15372:
15334:Mas-Colell, A.
15330:
15316:
15306:differentiable
15296:
15278:(3): 207–215.
15264:
15244:
15232:10.1086/258539
15226:(5): 478–479.
15212:
15206:
15186:
15172:
15159:
15132:
15100:
15086:
15065:
15051:
15035:
15021:
15005:
14991:
14966:
14954:10.1086/258544
14937:
14925:10.1086/258541
14919:(5): 484–489.
14908:
14896:10.1086/258197
14890:(4): 371–391.
14877:
14871:
14850:
14836:
14812:
14798:
14769:
14743:(3): 285–291.
14730:
14710:Debreu, Gérard
14706:
14669:
14655:
14639:
14621:(3): 433–436.
14607:
14581:
14566:
14538:Borwein, J. M.
14534:
14495:
14480:
14464:
14458:
14442:
14430:10.1086/258542
14413:
14401:10.1086/258540
14395:(5): 480–483.
14384:
14344:
14306:
14280:(1–2): 39–50.
14266:
14240:(3): 225–245.
14227:
14213:
14200:
14156:(5): 879–882.
14141:
14140:
14139:
14129:
14073:(2): 172–185.
14062:
14048:
14024:
13995:(March 1995).
13988:
13986:
13983:
13981:
13980:
13978:, p. 210)
13968:
13964:Artstein (1980
13956:
13906:
13902:Tardella (1990
13894:
13892:, p. 345)
13882:
13866:
13854:
13839:
13827:
13815:
13803:
13791:
13779:
13737:
13713:
13701:
13699:, p. 496)
13686:
13674:
13662:
13647:
13635:
13631:Ichiishi (1983
13623:
13611:
13604:
13577:
13568:, p. 169)
13539:
13521:
13510:
13508:, pp. 99)
13498:
13486:
13470:
13455:
13434:
13411:
13409:, p. 806)
13399:
13397:, p. 182)
13387:
13370:
13366:Koopmans (1957
13350:Farrell (1961a
13342:Koopmans (1961
13334:
13317:
13303:Farrell (
13296:
13284:
13266:
13264:, p. 146)
13250:
13238:
13226:
13214:
13198:
13196:, p. 138)
13183:
13164:(2): 371–372.
13144:
13125:(2): 314–317.
13106:
13099:
13078:
13048:
13046:, p. 180)
13044:Artstein (1980
13036:
13024:
13022:, p. 128)
13012:
12997:
12985:
12973:
12956:
12954:, p. 129)
12944:
12932:
12917:
12905:
12903:, p. 387)
12893:
12881:
12865:
12849:
12833:
12813:
12798:
12796:, p. 375)
12786:
12767:
12752:
12713:Ekeland (1999)
12698:
12673:
12657:
12642:
12621:
12619:
12618:
12551:
12540:
12539:
12535:can be strict
12508:
12456:
12454:
12451:
12435:control theory
12423:
12422:
12418:
12410:
12399:
12393:
12385:
12382:
12375:
12369:
12353:
12346:
12337:
12330:
12310:
12303:
12297:vector measure
12289:vector measure
12191:expected value
12185:) subset
12174:
12171:
12128:
12127:
12123:
12117:
12110:
12106:
12103:
12100:
12096:
12083:
12077:
12062:
12058:
12047:
12041:
12035:
12024:
12023:
12020:
12007:
11998:
11991:
11987: = (
11984:
11971:
11970:
11964:
11955:
11948:
11944:
11931:
11922:
11915:
11908:
11889:is the sum of
11874:
11871:
11849:function
11841:) =
11827:
11826:
11816:
11815:
11811:
11806:) ≤
11798:) :
11787:
11763:
11762:
11740:
11739:
11735:
11734:
11731:
11716:
11712:
11711:
11703:
11702:
11661:
11658:
11644:problems with
11613:
11610:
11578:microeconomics
11566:Leonid Hurwicz
11558:Paul Samuelson
11517:
11516:
11505:
11501:
11495:
11491:
11480:
11424:Nobel laureate
11412:
11409:
11368:Paul Samuelson
11312:
11309:
11275:preference set
11245:
11238:
11223:
11216:
11209:
11202:
11183:
11180:
11163:
11160:
11148:Leonid Hurwicz
11140:Paul Samuelson
11059:
11056:
11042:
11037:
11033:
11029:
11026:
11023:
11020:
11015:
11012:
11008:
11004:
11001:
10998:
10995:
10992:
10989:
10986:
10966:
10963:
10960:
10938:
10935:
10932:
10929:
10926:
10921:
10917:
10913:
10910:
10907:
10904:
10899:
10896:
10892:
10888:
10884:
10880:
10877:
10874:
10869:
10865:
10861:
10858:
10855:
10852:
10848:
10842:
10839:
10836:
10832:
10828:
10824:
10819:
10815:
10811:
10807:
10804:
10801:
10796:
10793:
10790:
10786:
10782:
10778:
10772:
10768:
10762:
10759:
10756:
10752:
10747:
10743:
10740:
10737:
10734:
10731:
10728:
10725:
10722:
10719:
10716:
10692:
10688:
10665:
10661:
10655:
10652:
10649:
10645:
10641:
10636:
10632:
10626:
10623:
10620:
10616:
10612:
10609:
10587:
10583:
10559:
10556:
10553:
10533:
10530:
10527:
10522:
10518:
10514:
10511:
10508:
10505:
10502:
10499:
10494:
10490:
10486:
10483:
10480:
10477:
10455:
10451:
10447:
10444:
10439:
10435:
10431:
10427:
10404:
10400:
10377:
10373:
10350:
10346:
10325:
10322:
10319:
10310:Now, for each
10294:
10290:
10286:
10281:
10277:
10273:
10270:
10265:
10261:
10257:
10253:
10250:
10247:
10244:
10240:
10235:
10231:
10227:
10224:
10221:
10217:
10213:
10209:
10205:
10202:
10199:
10196:
10193:
10190:
10187:
10184:
10181:
10178:
10175:
10172:
10152:
10149:
10146:
10124:
10120:
10114:
10111:
10108:
10104:
10100:
10095:
10091:
10085:
10082:
10079:
10075:
10071:
10068:
10048:
10045:
10042:
10038:
10035:
10032:
10029:
10025:
10022:
10019:
9997:
9992:
9988:
9984:
9981:
9978:
9975:
9970:
9967:
9963:
9959:
9955:
9952:
9949:
9945:
9942:
9939:
9925:
9911:
9908:
9905:
9902:
9899:
9896:
9893:
9890:
9887:
9884:
9881:
9859:
9856:
9853:
9850:
9847:
9844:
9841:
9838:
9835:
9832:
9827:
9823:
9819:
9816:
9813:
9810:
9807:
9804:
9801:
9798:
9795:
9792:
9789:
9786:
9783:
9780:
9777:
9774:
9771:
9768:
9765:
9762:
9759:
9756:
9736:
9716:
9713:
9710:
9707:
9704:
9701:
9698:
9695:
9675:
9655:
9652:
9649:
9646:
9643:
9640:
9620:
9600:
9580:
9577:
9574:
9570:
9567:
9564:
9561:
9557:
9554:
9534:
9531:
9528:
9525:
9522:
9519:
9516:
9513:
9510:
9507:
9504:
9501:
9498:
9476:
9473:
9470:
9467:
9464:
9461:
9458:
9453:
9449:
9445:
9442:
9439:
9435:
9432:
9429:
9426:
9422:
9419:
9416:
9413:
9400:
9387:
9384:
9381:
9378:
9375:
9372:
9369:
9366:
9363:
9360:
9357:
9354:
9351:
9348:
9345:
9342:
9339:
9336:
9333:
9328:
9324:
9320:
9317:
9314:
9310:
9307:
9304:
9301:
9297:
9294:
9291:
9288:
9268:
9265:
9262:
9259:
9256:
9253:
9248:
9240:
9237:
9234:
9231:
9228:
9225:
9222:
9218:
9213:
9207:
9204:
9201:
9197:
9194:
9191:
9188:
9184:
9181:
9177:
9173:
9170:
9167:
9164:
9161:
9158:
9155:
9135:
9112:
9109:
9106:
9102:
9098:
9095:
9073:
9069:
9065:
9062:
9057:
9053:
9049:
9046:
9043:
9040:
9037:
9017:
8997:
8977:
8974:
8969:
8965:
8942:
8938:
8932:
8928:
8924:
8921:
8918:
8898:
8895:
8892:
8888:
8885:
8882:
8879:
8875:
8872:
8869:
8849:
8846:
8843:
8839:
8836:
8833:
8830:
8800:
8796:
8773:
8770:
8767:
8760:
8757:
8749:
8745:
8742:
8739:
8735:
8729:
8725:
8704:
8684:
8663:
8657:
8654:
8651:
8644:
8641:
8633:
8629:
8626:
8623:
8619:
8613:
8609:
8604:
8598:
8594:
8590:
8584:
8581:
8552:
8548:
8545:
8542:
8538:
8517:
8514:
8511:
8508:
8505:
8485:
8464:
8460:
8457:
8454:
8450:
8429:
8426:
8423:
8402:
8398:
8395:
8392:
8388:
8367:
8344:
8340:
8336:
8333:
8328:
8324:
8319:
8313:
8309:
8305:
8302:
8281:
8277:
8274:
8269:
8265:
8260:
8256:
8249:
8245:
8241:
8238:
8234:
8228:
8225:
8220:
8216:
8208:
8205:
8198:
8194:
8190:
8187:
8182:
8178:
8173:
8150:
8146:
8142:
8139:
8119:
8112:
8108:
8104:
8101:
8097:
8091:
8088:
8085:
8078:
8075:
8068:
8064:
8060:
8057:
8054:
8050:
8044:
8040:
8034:
8030:
8026:
8003:
7999:
7995:
7992:
7987:
7981:
7978:
7975:
7968:
7965:
7957:
7953:
7950:
7947:
7943:
7937:
7933:
7927:
7923:
7918:
7913:
7906:
7902:
7898:
7895:
7891:
7884:
7881:
7875:
7852:
7849:
7826:
7803:
7800:
7777:
7772:
7768:
7764:
7761:
7758:
7734:
7730:
7726:
7723:
7719:
7712:
7709:
7703:
7682:
7678:
7675:
7670:
7666:
7661:
7640:
7620:
7617:
7614:
7611:
7606:
7602:
7578:
7558:
7553:
7549:
7545:
7540:
7537:
7534:
7530:
7526:
7523:
7518:
7515:
7512:
7505:
7502:
7494:
7491:
7488:
7485:
7482:
7479:
7476:
7473:
7470:
7467:
7464:
7461:
7458:
7452:
7449:
7424:
7421:
7418:
7409:, and at most
7398:
7395:
7391:
7387:
7384:
7381:
7377:
7354:
7351:
7348:
7341:
7338:
7330:
7326:
7323:
7320:
7316:
7310:
7306:
7300:
7296:
7292:
7286:
7283:
7253:
7250:
7247:
7244:
7241:
7238:
7235:
7232:
7229:
7226:
7223:
7219:
7213:
7210:
7207:
7200:
7197:
7190:
7167:
7164:
7141:
7136:
7133:
7130:
7123:
7120:
7111:
7108:
7105:
7101:
7095:
7091:
7085:
7081:
7077:
7071:
7068:
7040:
7018:
7013:
6989:
6985:
6964:
6959:
6955:
6951:
6946:
6943:
6940:
6936:
6932:
6929:
6924:
6921:
6918:
6911:
6908:
6900:
6897:
6894:
6891:
6888:
6885:
6882:
6879:
6876:
6873:
6870:
6867:
6864:
6858:
6855:
6830:
6827:
6824:
6819:
6795:
6790:
6766:
6763:
6760:
6756:
6750:
6747:
6744:
6740:
6734:
6730:
6724:
6720:
6716:
6713:
6690:
6687:
6682:
6679:
6676:
6672:
6666:
6662:
6641:
6638:
6633:
6630:
6627:
6623:
6602:
6580:
6577:
6574:
6570:
6564:
6561:
6558:
6554:
6548:
6543:
6540:
6537:
6533:
6529:
6524:
6520:
6499:
6494:
6490:
6486:
6482:
6479:
6476:
6473:
6469:
6464:
6460:
6439:
6425:
6410:
6405:
6377:
6374:
6371:
6366:
6342:
6337:
6315:
6303:
6300:
6260:
6257:
6244:
6239:
6233:
6228:
6224:
6220:
6214:
6210:
6204:
6200:
6194:
6189:
6186:
6183:
6179:
6173:
6170:
6165:
6161:
6155:
6151:
6145:
6140:
6137:
6134:
6130:
6124:
6121:
6115:
6110:
6107:
6104:
6101:
6096:
6090:
6086:
6063:
6059:
6036:
6032:
6011:
6008:
6003:
5999:
5976:
5971:
5947:
5944:
5941:
5938:
5935:
5932:
5929:
5926:
5923:
5919:
5913:
5909:
5905:
5879:
5875:
5869:
5865:
5861:
5858:
5853:
5850:
5846:
5842:
5839:
5836:
5833:
5830:
5826:
5823:
5820:
5817:
5813:
5810:
5807:
5802:
5798:
5781:
5766:
5761:
5756:
5753:
5733:
5730:
5727:
5724:
5721:
5718:
5715:
5712:
5709:
5706:
5703:
5679:
5676:
5655:
5634:
5631:
5627:
5624:
5602:
5597:
5592:
5589:
5586:
5582:
5579:
5546:
5543:
5540:
5537:
5534:
5530:
5527:
5524:
5521:
5517:
5514:
5494:
5474:
5454:
5451:
5448:
5444:
5441:
5438:
5435:
5431:
5428:
5408:
5386:
5381:
5376:
5373:
5353:
5350:
5347:
5344:
5329:
5326:
5306:
5284:
5281:
5277:
5253:
5248:
5244:
5238:
5234:
5230:
5227:
5224:
5221:
5216:
5213:
5209:
5202:
5196:
5193:
5190:
5187:
5183:
5180:
5177:
5174:
5170:
5167:
5164:
5159:
5155:
5138:
5125:
5122:
5119:
5116:
5113:
5108:
5105:
5102:
5098:
5090:
5086:
5082:
5079:
5075:
5071:
5068:
5065:
5062:
5059:
5056:
5053:
5029:
5026:
5023:
5020:
5017:
5014:
4991:
4986:
4981:
4976:
4973:
4951:
4948:
4945:
4942:
4939:
4936:
4933:
4929:
4926:
4923:
4920:
4916:
4911:
4907:
4886:
4863:
4858:
4854:
4850:
4847:
4844:
4841:
4838:
4835:
4832:
4829:
4826:
4823:
4820:
4817:
4814:
4811:
4808:
4805:
4802:
4798:
4794:
4791:
4788:
4785:
4782:
4779:
4776:
4773:
4770:
4767:
4764:
4761:
4758:
4755:
4752:
4749:
4746:
4741:
4737:
4714:
4709:
4705:
4700:
4697:
4693:
4690:
4687:
4684:
4681:
4678:
4674:
4671:
4667:
4664:
4661:
4656:
4652:
4631:
4628:
4625:
4622:
4619:
4616:
4611:
4608:
4605:
4601:
4597:
4594:
4591:
4588:
4585:
4582:
4579:
4559:
4554:
4550:
4546:
4543:
4540:
4537:
4534:
4531:
4528:
4525:
4522:
4519:
4516:
4511:
4507:
4486:
4481:
4477:
4473:
4470:
4467:
4464:
4461:
4456:
4453:
4450:
4446:
4442:
4439:
4436:
4433:
4430:
4427:
4424:
4419:
4415:
4392:
4387:
4383:
4379:
4376:
4373:
4370:
4364:
4361:
4357:
4354:
4351:
4348:
4345:
4342:
4338:
4334:
4331:
4327:
4324:
4320:
4317:
4314:
4309:
4305:
4284:
4279:
4274:
4269:
4265:
4262:
4258:
4255:
4235:
4230:
4226:
4222:
4219:
4216:
4213:
4208:
4205:
4202:
4198:
4194:
4191:
4188:
4185:
4182:
4179:
4174:
4170:
4142:
4137:
4132:
4127:
4124:
4115:and any point
4102:
4097:
4092:
4089:
4073:using squared
4062:
4059:
4056:
4052:
4049:
4046:
4043:
4022:
4010:
4007:
3990:
3970:
3950:
3933:
3931:
3928:
3927:
3926:
3915:
3911:
3905:
3901:
3895:
3892:
3889:
3885:
3881:
3878:
3873:
3869:
3865:
3861:
3858:
3855:
3852:
3846:
3843:
3840:
3836:
3831:
3825:
3822:
3818:
3814:
3810:
3803:
3800:
3797:
3794:
3791:
3788:
3785:
3782:
3779:
3776:
3773:
3769:
3765:
3761:
3755:
3751:
3745:
3740:
3737:
3734:
3730:
3725:
3720:
3717:
3714:
3711:
3684:
3664:
3661:
3658:
3655:
3652:
3649:
3646:
3624:
3620:
3616:
3611:
3607:
3586:
3583:
3580:
3577:
3574:
3554:
3549:
3545:
3541:
3537:
3534:
3531:
3528:
3524:
3519:
3515:
3503:
3502:
3489:
3485:
3479:
3474:
3471:
3468:
3465:
3462:
3458:
3454:
3449:
3445:
3439:
3434:
3431:
3428:
3424:
3420:
3417:
3394:
3391:
3388:
3384:
3381:
3378:
3375:
3351:
3348:
3345:
3342:
3339:
3319:
3316:
3313:
3310:
3307:
3287:
3284:
3281:
3278:
3275:
3272:
3269:
3265:
3262:
3259:
3256:
3252:
3249:
3246:
3243:
3240:
3237:
3234:
3231:
3227:
3224:
3221:
3218:
3214:
3211:
3208:
3205:
3202:
3199:
3196:
3193:
3190:
3187:
3184:
3181:
3178:
3175:
3172:
3169:
3166:
3163:
3137:
3133:
3129:
3124:
3120:
3097:
3093:
3089:
3086:
3081:
3077:
3073:
3069:
3066:
3063:
3060:
3056:
3051:
3047:
3026:
3017:, and at most
3006:
3003:
2998:
2994:
2988:
2983:
2980:
2977:
2973:
2950:
2946:
2925:
2922:
2919:
2915:
2912:
2909:
2906:
2902:
2899:
2882:
2865:
2862:
2857:
2853:
2847:
2842:
2839:
2836:
2832:
2811:
2806:
2802:
2798:
2794:
2791:
2788:
2785:
2781:
2776:
2772:
2751:
2748:
2745:
2741:
2738:
2735:
2732:
2728:
2725:
2705:
2700:
2696:
2692:
2688:
2685:
2682:
2679:
2673:
2668:
2665:
2662:
2658:
2654:
2651:
2648:
2645:
2641:
2638:
2635:
2632:
2619:
2616:
2603:
2600:
2597:
2593:
2590:
2587:
2584:
2563:
2539:
2535:
2529:
2524:
2521:
2518:
2514:
2510:
2507:
2485:
2463:
2458:
2434:
2430:
2426:
2423:
2420:
2417:
2414:
2409:
2405:
2380:
2375:
2353:
2331:
2328:
2325:
2322:
2319:
2316:
2313:
2310:
2307:
2304:
2301:
2298:
2295:
2292:
2289:
2278:
2275:
2273:
2270:
2257:
2254:
2251:
2248:
2245:
2225:
2222:
2217:
2213:
2191:
2187:
2184:
2173:
2172:
2161:
2156:
2152:
2148:
2144:
2141:
2138:
2135:
2129:
2124:
2121:
2118:
2114:
2110:
2107:
2102:
2098:
2092:
2087:
2084:
2081:
2077:
2073:
2069:
2066:
2063:
2060:
2045:
2044:
2033:
2030:
2027:
2024:
2020:
2017:
2014:
2011:
2007:
2004:
2001:
1998:
1994:
1991:
1988:
1985:
1981:
1978:
1975:
1972:
1969:
1966:
1962:
1959:
1956:
1953:
1925:
1905:
1902:
1899:
1896:
1893:
1881:
1878:
1877:
1876:
1865:
1862:
1859:
1856:
1853:
1850:
1847:
1841:
1836:
1832:
1828:
1823:
1819:
1815:
1810:
1806:
1800:
1795:
1792:
1789:
1785:
1781:
1778:
1773:
1769:
1763:
1758:
1755:
1752:
1748:
1724:
1719:
1715:
1711:
1708:
1705:
1700:
1696:
1692:
1687:
1683:
1679:
1674:
1670:
1664:
1659:
1656:
1653:
1649:
1637:
1636:
1625:
1622:
1619:
1616:
1613:
1610:
1607:
1604:
1601:
1598:
1595:
1592:
1589:
1586:
1583:
1580:
1577:
1574:
1571:
1568:
1565:
1562:
1559:
1556:
1553:
1550:
1547:
1544:
1541:
1538:
1535:
1532:
1529:
1526:
1523:
1520:
1517:
1494:
1491:
1488:
1485:
1482:
1476:
1473:
1470:
1467:
1464:
1461:
1458:
1455:
1452:
1449:
1446:
1443:
1440:
1420:
1417:
1414:
1411:
1408:
1382:
1356:
1352:
1348:
1345:
1342:
1339:
1336:
1331:
1327:
1323:
1318:
1314:
1304:is the square
1291:
1287:
1283:
1280:
1277:
1274:
1271:
1266:
1262:
1239:
1235:
1231:
1228:
1225:
1222:
1219:
1214:
1210:
1190:
1187:
1181:is the closed
1105:
1102:
1083:
1076:
1069:
1061:
1052:
1046:
1039:
1033:
1026:
1020:
1012:
1005:
998:
957:
956:
953:
946:
919:
912:
895:empty set
874:
853:
844:
827:
818:
777:
776:
758:non-convex set
755:
748:
747:
727:
720:
719:
718:
717:
716:
714:
711:
692:
685:
678:
661:
639:
614:
603:
602:
566:
565:
561:
554:
547:
540:
533:
526:
519:
512:
468:
465:
456:
453:
444:
443:
430:
429:
400:
397:
394:
391:
388:
385:
382:
379:
376:
373:
370:
367:
364:
361:
358:
355:
352:
349:
346:
343:
340:
337:
334:
331:
328:
325:
322:
319:
316:
313:
310:
307:
304:
301:
298:
295:
292:
289:
264:if and only if
254:of a set
236:
215:
206:
189:
180:
160:
157:
15:
9:
6:
4:
3:
2:
17706:
17695:
17692:
17690:
17687:
17685:
17684:Lloyd Shapley
17682:
17680:
17677:
17675:
17672:
17670:
17667:
17665:
17662:
17660:
17657:
17655:
17652:
17651:
17649:
17642:
17632:
17629:
17628:
17626:
17622:
17616:
17613:
17611:
17608:
17606:
17603:
17601:
17598:
17597:
17595:
17593:
17589:
17582:
17580:
17574:
17572:
17566:
17562:
17558:
17554:
17551:
17550:
17548:
17544:
17538:
17535:
17533:
17529:
17526:
17524:
17521:
17519:
17516:
17514:
17511:
17509:
17506:
17504:
17501:
17499:
17495:
17491:
17487:
17485:
17482:
17481:
17479:
17475:
17469:
17466:
17464:
17461:
17459:
17456:
17454:
17451:
17449:
17448:Mazur's lemma
17446:
17444:
17441:
17439:
17436:
17434:
17431:
17429:
17426:
17424:
17421:
17419:
17416:
17414:
17411:
17410:
17408:
17406:
17401:
17395:
17394:Subderivative
17392:
17390:
17387:
17385:
17382:
17380:
17377:
17375:
17371:
17368:
17366:
17363:
17361:
17358:
17356:
17353:
17351:
17348:
17346:
17342:
17340:
17337:
17335:
17332:
17331:
17329:
17325:
17319:
17316:
17314:
17311:
17309:
17306:
17304:
17301:
17299:
17296:
17294:
17291:
17289:
17286:
17284:
17281:
17279:
17276:
17275:
17273:
17271:
17270:Topics (list)
17267:
17261:
17258:
17256:
17253:
17251:
17248:
17247:
17245:
17241:
17237:
17233:
17226:
17221:
17219:
17214:
17212:
17207:
17206:
17203:
17191:
17183:
17181:
17176:
17171:
17170:
17167:
17161:
17158:
17156:
17153:
17149:
17146:
17145:
17144:
17141:
17140:
17138:
17134:
17128:
17125:
17123:
17120:
17118:
17115:
17113:
17110:
17108:
17105:
17103:
17100:
17098:
17095:
17093:
17090:
17088:
17087:Institutional
17085:
17083:
17080:
17078:
17075:
17073:
17070:
17068:
17065:
17063:
17060:
17058:
17055:
17053:
17050:
17048:
17045:
17043:
17040:
17038:
17035:
17033:
17032:Computational
17030:
17028:
17025:
17023:
17020:
17019:
17017:
17013:
17007:
17004:
17000:
16997:
16995:
16992:
16991:
16990:
16987:
16985:
16982:
16978:
16977:Law of supply
16974:
16971:
16969:
16968:Law of demand
16965:
16962:
16961:
16960:
16957:
16955:
16954:Social choice
16952:
16950:
16947:
16945:
16942:
16940:
16939:Excess supply
16936:
16933:
16931:
16928:
16926:
16925:Risk aversion
16923:
16921:
16918:
16916:
16913:
16911:
16908:
16906:
16903:
16901:
16898:
16896:
16893:
16889:
16886:
16884:
16880:
16877:
16875:
16872:
16870:
16867:
16865:
16862:
16860:
16859:Price ceiling
16857:
16855:
16852:
16851:
16850:
16847:
16845:
16842:
16840:
16837:
16833:
16830:
16828:
16825:
16823:
16820:
16816:
16815:Complementary
16813:
16811:
16808:
16807:
16806:
16803:
16801:
16798:
16794:
16791:
16789:
16786:
16785:
16784:
16781:
16780:
16779:
16776:
16774:
16771:
16769:
16766:
16764:
16761:
16759:
16756:
16754:
16751:
16749:
16746:
16744:
16741:
16737:
16734:
16732:
16729:
16728:
16727:
16724:
16722:
16719:
16717:
16714:
16712:
16709:
16705:
16702:
16701:
16700:
16697:
16693:
16690:
16688:
16685:
16683:
16680:
16678:
16675:
16674:
16673:
16670:
16668:
16665:
16663:
16660:
16658:
16655:
16653:
16650:
16648:
16645:
16641:
16638:
16636:
16633:
16631:
16628:
16624:
16621:
16620:
16619:
16616:
16614:
16611:
16609:
16606:
16605:
16604:
16601:
16599:
16598:non-convexity
16595:
16592:
16590:
16587:
16585:
16582:
16580:
16577:
16576:
16574:
16570:
16566:
16559:
16554:
16552:
16547:
16545:
16540:
16539:
16536:
16524:
16516:
16515:
16512:
16506:
16503:
16501:
16498:
16494:
16491:
16490:
16489:
16486:
16485:
16483:
16479:
16473:
16470:
16468:
16465:
16463:
16460:
16458:
16455:
16454:
16452:
16448:
16442:
16439:
16437:
16434:
16432:
16429:
16427:
16424:
16422:
16419:
16417:
16414:
16412:
16409:
16408:
16406:
16404:
16398:
16392:
16389:
16385:
16382:
16380:
16377:
16376:
16375:
16372:
16370:
16367:
16365:
16362:
16360:
16359:Combinatorial
16357:
16356:
16354:
16352:
16346:
16340:
16337:
16333:
16330:
16329:
16328:
16325:
16324:
16321:
16317:
16310:
16305:
16303:
16298:
16296:
16291:
16290:
16287:
16271:
16264:
16260:
16256:
16241:
16237:
16233:
16229:
16227:9781139174749
16223:
16219:
16215:
16211:
16204:
16202:
16196:
16192:
16180:
16173:
16168:
16167:
16156:
16152:
16148:
16144:
16140:
16136:
16132:
16128:
16124:
16120:
16116:
16112:
16108:
16104:
16100:
16096:
16092:
16087:
16082:
16078:
16074:
16070:
16066:
16061:
16057:
16053:
16049:
16045:
16041:
16037:
16032:
16028:
16024:
16020:
16014:
16009:
16008:
16002:
15998:
15994:
15990:
15986:
15984:3-540-12881-6
15980:
15976:
15971:
15967:
15963:
15959:
15955:
15951:
15947:
15942:
15938:
15932:
15928:
15924:
15920:
15916:
15912:
15908:
15904:
15900:
15896:
15894:0-521-56473-5
15890:
15886:
15882:
15877:
15873:
15867:
15863:
15859:
15855:
15854:
15849:
15845:
15841:
15836:
15832:
15828:
15824:
15820:
15816:
15812:
15808:
15804:
15800:
15796:
15792:
15788:
15784:
15780:
15776:
15772:
15768:
15764:
15760:
15756:
15752:
15748:
15744:
15740:
15734:
15730:
15726:
15722:
15717:
15713:
15709:
15705:
15703:0-521-35220-7
15699:
15695:
15694:
15688:
15684:
15680:
15676:
15672:
15668:
15664:
15660:
15656:
15652:
15648:
15645:
15639:
15637:0-262-19443-0
15633:
15629:
15624:
15620:
15616:
15611:
15606:
15602:
15598:
15597:
15592:
15588:
15584:
15580:
15576:
15572:
15568:
15564:
15560:
15555:
15551:
15547:
15543:
15539:
15535:
15531:
15526:
15524:
15520:
15517:
15511:
15507:
15503:
15501:0-691-01586-4
15497:
15493:
15489:
15485:
15481:
15477:
15473:
15469:
15465:
15461:
15457:
15453:
15448:
15444:
15440:
15436:
15430:
15426:
15422:
15418:
15413:
15412:
15405:
15401:
15395:
15391:
15387:
15383:
15379:
15375:
15373:9780333786765
15369:
15365:
15361:
15357:
15356:
15351:
15350:Newman, Peter
15347:
15343:
15342:Eatwell, John
15339:
15335:
15331:
15327:
15323:
15319:
15317:0-521-26514-2
15313:
15309:
15305:
15301:
15297:
15293:
15289:
15285:
15281:
15277:
15273:
15269:
15265:
15261:
15257:
15253:
15249:
15245:
15241:
15237:
15233:
15229:
15225:
15221:
15217:
15213:
15209:
15207:0-07-035337-9
15203:
15199:
15195:
15191:
15187:
15183:
15179:
15175:
15173:0-12-370180-5
15169:
15165:
15160:
15148:
15141:
15137:
15133:
15129:
15125:
15121:
15117:
15113:
15109:
15105:
15101:
15097:
15093:
15089:
15087:3-540-56852-2
15083:
15079:
15077:
15071:
15066:
15062:
15058:
15054:
15048:
15044:
15040:
15036:
15032:
15028:
15024:
15022:0-262-03149-3
15018:
15014:
15010:
15006:
15002:
14998:
14994:
14992:0-444-86126-2
14988:
14984:
14980:
14976:
14972:
14967:
14963:
14959:
14955:
14951:
14947:
14943:
14938:
14934:
14930:
14926:
14922:
14918:
14914:
14909:
14905:
14901:
14897:
14893:
14889:
14885:
14884:
14878:
14874:
14868:
14864:
14860:
14856:
14851:
14847:
14843:
14839:
14837:0-89871-450-8
14833:
14829:
14825:
14821:
14820:a priori
14817:
14816:Ekeland, Ivar
14813:
14809:
14805:
14801:
14795:
14791:
14787:
14783:
14779:
14775:
14770:
14766:
14762:
14758:
14754:
14750:
14746:
14742:
14738:
14737:
14731:
14727:
14723:
14719:
14715:
14711:
14707:
14703:
14699:
14694:
14689:
14685:
14681:
14680:
14675:
14670:
14666:
14662:
14658:
14656:0-521-28614-X
14652:
14648:
14644:
14640:
14636:
14632:
14628:
14624:
14620:
14616:
14612:
14608:
14604:
14600:
14596:
14592:
14588:
14584:
14582:0-262-53192-5
14578:
14574:
14573:
14567:
14563:
14559:
14555:
14551:
14547:
14543:
14539:
14535:
14523:
14519:
14515:
14511:
14504:
14500:
14496:
14491:
14487:
14483:
14481:1-886529-04-3
14477:
14473:
14469:
14465:
14461:
14459:1-886529-00-0
14455:
14451:
14447:
14443:
14439:
14435:
14431:
14427:
14423:
14419:
14414:
14410:
14406:
14402:
14398:
14394:
14390:
14385:
14381:
14377:
14373:
14369:
14365:
14361:
14357:
14353:
14349:
14345:
14341:
14337:
14332:
14327:
14323:
14319:
14315:
14311:
14307:
14303:
14299:
14295:
14291:
14287:
14283:
14279:
14275:
14271:
14267:
14263:
14259:
14255:
14251:
14247:
14243:
14239:
14235:
14234:
14228:
14224:
14220:
14216:
14210:
14206:
14201:
14197:
14193:
14189:
14185:
14181:
14177:
14173:
14169:
14164:
14159:
14155:
14151:
14147:
14142:
14136:
14132:
14130:0-472-10673-2
14126:
14122:
14118:
14113:
14112:
14110:
14106:
14102:
14096:
14092:
14088:
14084:
14080:
14076:
14072:
14068:
14063:
14059:
14055:
14051:
14049:0-444-85497-5
14045:
14041:
14037:
14033:
14029:
14025:
14013:
14009:
14002:
14000:
13994:
13993:Aardal, Karen
13990:
13989:
13977:
13972:
13965:
13960:
13953:
13951:
13947:
13943:
13939:
13935:
13927:
13923:
13922:Gérard Debreu
13919:
13915:
13910:
13903:
13898:
13891:
13886:
13879:
13875:
13870:
13863:
13858:
13851:
13846:
13844:
13836:
13831:
13824:
13823:Cassels (1975
13819:
13813:, p. 45)
13812:
13807:
13800:
13795:
13788:
13783:
13775:
13771:
13767:
13763:
13759:
13755:
13751:
13750:a priori
13744:
13742:
13735:
13733:
13729:
13727:
13722:
13717:
13711:, p. 23)
13710:
13705:
13698:
13693:
13691:
13684:, p. 30)
13683:
13682:Trockel (1984
13678:
13671:
13666:
13659:
13654:
13652:
13644:
13643:Cassels (1981
13639:
13632:
13627:
13620:
13619:Salanié (2000
13615:
13607:
13605:0-262-12127-1
13601:
13597:
13593:
13592:
13587:
13581:
13575:
13573:
13569:
13567:
13563:
13561:
13557:
13555:
13551:
13548:
13543:
13537:
13535:
13530:
13525:
13519:
13514:
13507:
13502:
13496:, p. 44)
13495:
13490:
13483:
13479:
13474:
13467:
13462:
13460:
13452:
13451:Diewert (1982
13448:
13444:
13438:
13431:
13427:
13423:
13418:
13416:
13408:
13403:
13396:
13391:
13384:
13380:
13374:
13367:
13363:
13359:
13355:
13351:
13347:
13346:Farrell (1959
13343:
13338:
13331:
13327:
13321:
13314:
13310:
13306:
13300:
13293:
13292:Diewert (1982
13288:
13281:
13275:
13270:
13263:
13259:
13254:
13248:, p. 74)
13247:
13242:
13235:
13230:
13223:
13218:
13211:
13207:
13202:
13195:
13190:
13188:
13179:
13175:
13171:
13167:
13163:
13159:
13155:
13148:
13140:
13136:
13132:
13128:
13124:
13120:
13116:
13110:
13102:
13096:
13092:
13088:
13082:
13066:
13059:
13052:
13045:
13040:
13033:
13032:Ekeland (1999
13028:
13021:
13016:
13009:
13008:Cassels (1975
13004:
13002:
12994:
12989:
12982:
12977:
12971:, p. 37)
12970:
12965:
12963:
12961:
12953:
12948:
12942:, p. 36)
12941:
12936:
12930:
12926:
12921:
12914:
12909:
12902:
12897:
12890:
12885:
12879:, p. 78)
12878:
12874:
12869:
12863:, p. 16)
12862:
12858:
12853:
12846:
12842:
12837:
12831:, p. 37)
12830:
12826:
12822:
12817:
12811:, p. 10)
12810:
12805:
12803:
12795:
12790:
12784:, p. 94)
12783:
12778:
12776:
12774:
12772:
12764:
12759:
12757:
12750:
12749:
12745:
12743:
12739:
12735:
12731:
12727:
12723:
12718:
12714:
12710:
12705:
12703:
12695:
12691:
12690:Ekeland (1999
12686:
12684:
12682:
12680:
12678:
12671:
12666:
12664:
12662:
12654:
12649:
12647:
12640:
12635:
12633:
12631:
12629:
12627:
12622:
12615:
12612:
12608:
12605:
12602:
12598:
12594:
12593:
12588:
12584:
12581:
12578:
12574:
12570:
12569:
12564:
12563:Ekeland (1999
12560:
12555:
12548:
12544:
12537:
12536:
12534:
12530:
12526:
12522:
12518:
12512:
12505:
12501:
12497:
12492:
12484:
12481:
12477:
12476:Serbonian Bog
12473:
12472:
12471:Paradise Lost
12467:
12461:
12457:
12450:
12448:
12444:
12440:
12436:
12432:
12428:
12416:
12409:
12405:
12398:
12391:
12381:
12374:
12368:
12367:
12366:
12364:
12360:
12352:
12345:
12336:
12329:
12324:
12320:
12316:
12309:
12302:
12298:
12294:
12290:
12286:
12282:
12278:
12274:
12270:
12266:
12262:
12258:
12254:
12249:
12247:
12243:
12239:
12235:
12231:
12227:
12219:
12215:
12211:
12207:
12203:
12199:
12196:
12192:
12188:
12184:
12180:
12170:
12167:
12163:
12159:
12155:
12151:
12147:
12144:
12140:
12136:
12131:
12120:
12116:
12104: ∈
12095:
12091:
12086:
12082:
12076:
12075:
12074:
12072:
12068:
12057:
12053:
12046:
12034:
12030:
12005:
11997:
11990:
11983:
11980:
11979:
11978:
11976:
11967:
11963:
11958:
11954:
11929:
11921:
11914:
11907:
11903:
11899:
11896:
11895:
11894:
11892:
11888:
11884:
11880:
11870:
11868:
11864:
11860:
11856:
11852:
11848:
11844:
11840:
11836:
11832:
11824:
11823:
11818:
11817:
11809:
11805:
11801:
11797:
11793:
11785:
11781:
11780:
11776:
11772:
11767:
11760:
11756:
11752:
11748:
11747:
11742:
11741:
11729:
11725:
11721:
11709:
11705:
11704:
11700:
11696:
11692:
11688:
11684:
11680:
11679:
11674:
11673:
11672:
11670:
11666:
11657:
11655:
11651:
11647:
11643:
11635:
11631:
11627:
11623:
11618:
11609:
11607:
11603:
11599:
11595:
11591:
11587:
11583:
11579:
11575:
11571:
11567:
11563:
11559:
11555:
11551:
11547:
11546:Gérard Debreu
11543:
11542:Robert Aumann
11539:
11535:
11530:
11524:
11520:
11514:
11510:
11506:
11500:
11496:
11490:
11486:
11485:
11484:
11479:
11475:
11470:
11468:
11464:
11460:
11459:Lloyd Shapley
11456:
11452:
11448:
11447:Kenneth Arrow
11444:
11436:
11433:
11429:
11425:
11421:
11420:Kenneth Arrow
11417:
11408:
11406:
11405:Robert Aumann
11402:
11398:
11397:Martin Shubik
11394:
11393:Lloyd Shapley
11390:
11386:
11382:
11378:
11377:
11371:
11369:
11366:and again by
11365:
11359:
11354:
11352:
11348:
11343:
11341:
11336:
11332:
11326:
11317:
11308:
11306:
11305:
11300:
11295:
11290:
11288:
11284:
11280:
11276:
11271:
11270:
11265:
11261:
11255:
11244:
11237:
11233:
11229:
11228:shown in blue
11222:
11215:
11208:
11201:
11197:
11193:
11190:The consumer
11188:
11179:
11177:
11173:
11169:
11159:
11157:
11153:
11149:
11145:
11141:
11137:
11133:
11129:
11128:Gérard Debreu
11125:
11124:Robert Aumann
11121:
11117:
11114:The topic of
11112:
11110:
11104:
11101:
11097:
11096:Kenneth Arrow
11093:
11089:
11085:
11081:
11080:Lloyd Shapley
11078:The lemma of
11073:
11069:
11068:Lloyd Shapley
11064:
11054:
11035:
11031:
11024:
11021:
11018:
11013:
11006:
11002:
10996:
10990:
10987:
10984:
10964:
10961:
10958:
10949:
10936:
10933:
10930:
10927:
10919:
10915:
10908:
10905:
10902:
10897:
10890:
10886:
10882:
10878:
10875:
10867:
10863:
10856:
10853:
10850:
10846:
10840:
10837:
10834:
10830:
10826:
10822:
10817:
10813:
10809:
10805:
10802:
10799:
10794:
10791:
10788:
10784:
10780:
10776:
10770:
10766:
10760:
10757:
10754:
10750:
10745:
10741:
10738:
10735:
10732:
10726:
10720:
10717:
10714:
10706:
10690:
10686:
10678:. Since each
10663:
10659:
10653:
10650:
10647:
10643:
10639:
10634:
10630:
10624:
10621:
10618:
10614:
10610:
10607:
10585:
10581:
10573:Let all such
10571:
10557:
10554:
10551:
10531:
10528:
10520:
10516:
10509:
10506:
10503:
10500:
10492:
10488:
10481:
10478:
10475:
10453:
10449:
10445:
10437:
10433:
10402:
10398:
10375:
10371:
10348:
10344:
10323:
10320:
10317:
10308:
10292:
10288:
10284:
10279:
10275:
10271:
10263:
10259:
10238:
10233:
10229:
10225:
10222:
10219:
10211:
10203:
10197:
10194:
10191:
10188:
10185:
10182:
10179:
10176:
10173:
10150:
10147:
10144:
10122:
10118:
10112:
10109:
10106:
10102:
10098:
10093:
10089:
10083:
10080:
10077:
10073:
10069:
10066:
10043:
10023:
10020:
10009:
9990:
9986:
9979:
9976:
9973:
9968:
9961:
9957:
9953:
9950:
9947:
9943:
9940:
9937:
9923:
9906:
9900:
9897:
9891:
9885:
9882:
9879:
9871:
9857:
9854:
9848:
9842:
9839:
9836:
9833:
9825:
9817:
9814:
9811:
9802:
9799:
9793:
9790:
9787:
9781:
9778:
9775:
9772:
9766:
9760:
9757:
9754:
9734:
9714:
9711:
9705:
9699:
9696:
9693:
9673:
9653:
9650:
9644:
9638:
9618:
9598:
9575:
9555:
9552:
9529:
9523:
9520:
9517:
9514:
9508:
9502:
9499:
9496:
9488:
9471:
9465:
9462:
9459:
9456:
9451:
9440:
9420:
9417:
9411:
9399:
9382:
9376:
9373:
9370:
9367:
9361:
9355:
9352:
9346:
9340:
9337:
9334:
9331:
9326:
9315:
9295:
9292:
9286:
9263:
9257:
9254:
9251:
9246:
9238:
9235:
9232:
9229:
9223:
9202:
9182:
9179:
9171:
9165:
9159:
9156:
9153:
9133:
9124:
9107:
9096:
9093:
9071:
9067:
9063:
9055:
9051:
9047:
9044:
9038:
9035:
9015:
8995:
8975:
8972:
8967:
8963:
8940:
8936:
8930:
8926:
8922:
8919:
8916:
8893:
8873:
8870:
8844:
8818:
8814:
8798:
8794:
8771:
8768:
8765:
8755:
8747:
8743:
8740:
8737:
8733:
8727:
8723:
8702:
8682:
8661:
8655:
8652:
8649:
8639:
8631:
8627:
8624:
8621:
8617:
8611:
8607:
8602:
8596:
8592:
8588:
8579:
8568:
8565:
8550:
8546:
8543:
8540:
8536:
8515:
8512:
8509:
8506:
8503:
8483:
8462:
8458:
8455:
8452:
8448:
8427:
8424:
8421:
8400:
8396:
8393:
8390:
8386:
8365:
8356:
8342:
8338:
8334:
8331:
8326:
8322:
8317:
8311:
8307:
8303:
8300:
8279:
8275:
8272:
8267:
8263:
8258:
8254:
8247:
8243:
8239:
8236:
8226:
8223:
8218:
8214:
8203:
8192:
8188:
8185:
8180:
8176:
8171:
8148:
8144:
8140:
8137:
8117:
8110:
8106:
8102:
8099:
8089:
8086:
8083:
8073:
8062:
8058:
8055:
8052:
8048:
8042:
8038:
8032:
8028:
8024:
8001:
7997:
7993:
7990:
7985:
7979:
7976:
7973:
7963:
7955:
7951:
7948:
7945:
7941:
7935:
7931:
7925:
7921:
7916:
7911:
7904:
7900:
7896:
7893:
7879:
7847:
7824:
7798:
7770:
7766:
7762:
7759:
7732:
7728:
7724:
7721:
7707:
7680:
7676:
7673:
7668:
7664:
7659:
7638:
7618:
7615:
7612:
7609:
7604:
7600:
7590:
7576:
7551:
7547:
7543:
7538:
7535:
7532:
7528:
7521:
7516:
7513:
7510:
7500:
7492:
7486:
7483:
7480:
7477:
7474:
7471:
7468:
7465:
7462:
7456:
7447:
7436:
7422:
7419:
7416:
7396:
7393:
7389:
7385:
7382:
7379:
7375:
7352:
7349:
7346:
7336:
7328:
7324:
7321:
7318:
7314:
7308:
7304:
7298:
7294:
7290:
7281:
7270:
7267:
7251:
7248:
7245:
7242:
7239:
7236:
7233:
7230:
7227:
7224:
7221:
7211:
7208:
7205:
7195:
7162:
7139:
7134:
7131:
7128:
7118:
7109:
7106:
7103:
7099:
7093:
7089:
7083:
7079:
7075:
7066:
7055:
7052:
7038:
7016:
6987:
6983:
6957:
6953:
6949:
6944:
6941:
6938:
6934:
6927:
6922:
6919:
6916:
6906:
6898:
6892:
6889:
6886:
6883:
6880:
6877:
6874:
6871:
6868:
6862:
6853:
6828:
6825:
6822:
6793:
6764:
6761:
6758:
6754:
6748:
6745:
6742:
6738:
6732:
6728:
6722:
6718:
6714:
6711:
6702:
6688:
6685:
6680:
6677:
6674:
6670:
6664:
6660:
6639:
6636:
6631:
6628:
6625:
6621:
6600:
6578:
6575:
6572:
6568:
6562:
6559:
6556:
6552:
6546:
6541:
6538:
6535:
6531:
6527:
6522:
6518:
6492:
6488:
6467:
6462:
6458:
6437:
6424:
6408:
6393:
6375:
6372:
6369:
6340:
6313:
6299:
6297:
6293:
6289:
6284:
6282:
6278:
6274:
6270:
6266:
6256:
6242:
6237:
6231:
6226:
6222:
6218:
6212:
6208:
6202:
6198:
6192:
6187:
6184:
6181:
6177:
6171:
6168:
6163:
6159:
6153:
6149:
6143:
6138:
6135:
6132:
6128:
6122:
6119:
6113:
6094:
6088:
6084:
6061:
6057:
6034:
6030:
6009:
6006:
6001:
5997:
5974:
5945:
5942:
5939:
5936:
5933:
5930:
5927:
5924:
5921:
5911:
5907:
5893:
5877:
5867:
5863:
5856:
5851:
5844:
5840:
5831:
5811:
5808:
5800:
5796:
5780:
5764:
5754:
5751:
5728:
5722:
5719:
5716:
5713:
5707:
5701:
5692:
5677:
5674:
5653:
5632:
5629:
5622:
5600:
5590:
5587:
5584:
5580:
5577:
5562:
5558:
5541:
5538:
5535:
5515:
5512:
5492:
5472:
5449:
5429:
5426:
5406:
5384:
5374:
5371:
5348:
5342:
5335:
5325:
5323:
5318:
5304:
5282:
5275:
5264:
5251:
5246:
5236:
5232:
5225:
5222:
5219:
5214:
5207:
5200:
5188:
5168:
5165:
5157:
5153:
5137:
5120:
5117:
5114:
5106:
5103:
5100:
5088:
5084:
5080:
5077:
5069:
5063:
5057:
5054:
5051:
5043:
5024:
5018:
5015:
5012:
5005:
4989:
4984:
4974:
4971:
4962:
4949:
4943:
4940:
4934:
4909:
4905:
4884:
4874:
4861:
4856:
4845:
4842:
4839:
4836:
4833:
4827:
4821:
4818:
4815:
4806:
4803:
4800:
4796:
4792:
4789:
4780:
4777:
4774:
4771:
4768:
4762:
4756:
4753:
4750:
4739:
4735:
4727:For example,
4725:
4712:
4707:
4698:
4695:
4691:
4688:
4682:
4679:
4672:
4669:
4665:
4662:
4654:
4650:
4629:
4623:
4620:
4617:
4609:
4606:
4603:
4595:
4589:
4586:
4583:
4577:
4557:
4552:
4544:
4541:
4538:
4532:
4529:
4523:
4520:
4517:
4509:
4505:
4484:
4479:
4468:
4465:
4462:
4454:
4451:
4448:
4437:
4431:
4428:
4425:
4417:
4413:
4403:
4390:
4385:
4377:
4374:
4371:
4362:
4359:
4355:
4352:
4349:
4346:
4343:
4340:
4332:
4325:
4322:
4318:
4315:
4307:
4303:
4282:
4277:
4267:
4263:
4260:
4256:
4253:
4233:
4228:
4220:
4217:
4214:
4206:
4203:
4200:
4192:
4186:
4183:
4180:
4172:
4168:
4160:
4156:
4153:define their
4140:
4135:
4125:
4122:
4100:
4090:
4087:
4078:
4076:
4057:
4020:
4006:
4002:
3988:
3968:
3948:
3913:
3909:
3903:
3899:
3893:
3890:
3887:
3883:
3879:
3871:
3867:
3844:
3841:
3838:
3834:
3829:
3823:
3820:
3812:
3801:
3795:
3792:
3789:
3786:
3783:
3780:
3774:
3771:
3767:
3763:
3759:
3753:
3749:
3743:
3738:
3735:
3732:
3728:
3723:
3701:
3700:
3699:
3696:
3682:
3662:
3659:
3656:
3653:
3650:
3647:
3644:
3622:
3618:
3614:
3609:
3605:
3584:
3581:
3578:
3575:
3572:
3547:
3543:
3522:
3517:
3513:
3487:
3483:
3477:
3472:
3469:
3466:
3463:
3460:
3456:
3452:
3447:
3443:
3437:
3432:
3429:
3426:
3422:
3418:
3415:
3408:
3407:
3406:
3389:
3363:
3346:
3343:
3340:
3314:
3311:
3308:
3279:
3276:
3273:
3250:
3241:
3238:
3235:
3212:
3206:
3203:
3200:
3194:
3188:
3185:
3182:
3176:
3170:
3167:
3164:
3151:
3135:
3131:
3127:
3122:
3118:
3095:
3091:
3079:
3075:
3054:
3049:
3045:
3024:
3004:
3001:
2996:
2992:
2986:
2981:
2978:
2975:
2971:
2948:
2944:
2920:
2900:
2897:
2881:
2879:
2863:
2860:
2855:
2851:
2845:
2840:
2837:
2834:
2830:
2804:
2800:
2779:
2774:
2770:
2746:
2726:
2723:
2698:
2694:
2671:
2666:
2663:
2660:
2656:
2652:
2646:
2615:
2598:
2561:
2553:
2537:
2533:
2527:
2522:
2519:
2516:
2512:
2508:
2505:
2497:
2483:
2461:
2432:
2428:
2424:
2421:
2418:
2415:
2412:
2407:
2403:
2394:
2378:
2351:
2343:
2326:
2323:
2320:
2317:
2314:
2311:
2308:
2305:
2302:
2296:
2293:
2290:
2287:
2269:
2255:
2252:
2249:
2246:
2243:
2223:
2220:
2215:
2211:
2185:
2182:
2154:
2150:
2127:
2122:
2119:
2116:
2112:
2108:
2100:
2096:
2090:
2085:
2082:
2079:
2075:
2050:
2049:
2048:
2031:
2025:
2005:
1999:
1979:
1973:
1970:
1967:
1943:
1942:
1941:
1939:
1923:
1903:
1900:
1897:
1894:
1891:
1863:
1857:
1854:
1851:
1848:
1845:
1839:
1834:
1830:
1826:
1821:
1817:
1813:
1808:
1804:
1798:
1793:
1790:
1787:
1783:
1776:
1771:
1767:
1761:
1756:
1753:
1750:
1746:
1738:
1737:
1736:
1722:
1717:
1713:
1709:
1706:
1703:
1698:
1694:
1690:
1685:
1681:
1677:
1672:
1668:
1662:
1657:
1654:
1651:
1647:
1620:
1617:
1614:
1611:
1608:
1602:
1596:
1593:
1590:
1587:
1584:
1581:
1578:
1575:
1572:
1569:
1566:
1563:
1560:
1557:
1554:
1548:
1542:
1539:
1536:
1530:
1524:
1521:
1518:
1508:
1507:
1506:
1492:
1486:
1483:
1480:
1474:
1471:
1468:
1465:
1462:
1459:
1456:
1453:
1447:
1444:
1441:
1438:
1418:
1415:
1412:
1409:
1406:
1398:
1397:
1396:Minkowski sum
1380:
1354:
1346:
1343:
1340:
1334:
1329:
1325:
1321:
1316:
1312:
1289:
1281:
1278:
1275:
1269:
1264:
1260:
1237:
1229:
1226:
1223:
1217:
1212:
1208:
1199:
1195:
1186:
1184:
1180:
1176:
1172:
1168:
1164:
1160:
1156:
1152:
1148:
1144:
1138:
1134:
1133:
1127:
1119:
1115:
1110:
1101:
1099:
1098:disjoint sets
1095:
1094:
1088:
1082:
1075:
1068:
1060:
1051:
1045:
1038:
1032:
1025:
1019:
1011:
1004:
997:
993:
989:
985:
981:
977:
974:, a set
973:
968:
966:
962:
952:
945:
941:
937:
936:
935:
933:
932:unit interval
929:
925:
918:
911:
907:
902:
900:
896:
892:
888:
872:
851:
842:
825:
816:
812:
808:
804:
803:
798:
794:
784:
780:
779:Line segments
771:
767:
763:
759:
752:
743:
739:
735:
731:
724:
710:
708:
704:
701:are defined:
700:
691:
684:
677:
659:
651:
637:
628:
612:
600:
596:
592:
588:
584:
581:
580:
579:
577:
573:
572:
560:
553:
546:
539:
532:
525:
518:
511:
507:
506:
505:
503:
502:
497:
493:
489:
485:
481:
477:
474:
464:
462:
455:Preliminaries
452:
450:
441:
440:
439:
437:
436:
427:
426:
425:
422:
420:
416:
411:
398:
392:
389:
386:
383:
380:
374:
368:
365:
362:
359:
356:
353:
350:
347:
344:
341:
338:
335:
332:
329:
326:
320:
314:
311:
308:
302:
296:
293:
290:
279:
275:
271:
267:
265:
261:
257:
253:
248:
234:
213:
204:
187:
178:
174:
170:
166:
156:
154:
150:
146:
142:
138:
134:
130:
125:
123:
122:
116:
112:
108:
104:
100:
99:
93:
88:
86:
81:
79:
75:
71:
70:Lloyd Shapley
67:
63:
59:
55:
51:
50:
40:
36:
32:
28:
23:
19:
17659:Convex hulls
17641:
17615:Weak duality
17578:
17570:
17490:Orthogonally
17452:
17122:Optimization
17107:Mathematical
17067:Experimental
17062:Evolutionary
17047:Econometrics
16905:Public goods
16879:Price system
16874:Price signal
16788:Monopolistic
16657:Distribution
16572:Major topics
16457:Trigonometry
16274:. Retrieved
16269:
16261:(May 2007).
16247:. Retrieved
16240:the original
16209:
16200:
16183:. Retrieved
16178:
16146:
16143:Wold, Herman
16118:
16114:
16110:
16106:
16103:Wold, Herman
16076:
16072:
16068:
16039:
16035:
16006:
15974:
15949:
15945:
15918:
15884:
15880:
15852:
15817:(1): 25–38.
15814:
15811:Econometrica
15810:
15799:the original
15770:
15767:Econometrica
15766:
15720:
15692:
15658:
15654:
15643:
15627:
15600:
15594:
15562:
15558:
15533:
15529:
15522:
15491:
15455:
15451:
15410:
15389:
15354:
15307:
15303:
15275:
15271:
15252:Rocquencourt
15223:
15219:
15197:
15163:
15151:. Retrieved
15114:(1): 66–78.
15111:
15108:Econometrica
15107:
15075:
15073:
15042:
15012:
14974:
14945:
14941:
14916:
14912:
14887:
14881:
14854:
14827:
14824:Temam, Roger
14819:
14781:
14740:
14734:
14717:
14713:
14683:
14677:
14646:
14618:
14614:
14603:the original
14571:
14545:
14541:
14525:. Retrieved
14513:
14509:
14471:
14449:
14421:
14417:
14392:
14388:
14355:
14352:Econometrica
14351:
14321:
14317:
14277:
14274:Econometrica
14273:
14237:
14231:
14204:
14153:
14149:
14135:the original
14120:
14070:
14066:
14039:
14035:
14016:. Retrieved
14011:
14007:
13998:
13971:
13959:
13949:
13945:
13941:
13937:
13933:
13930:
13926:Debreu (1991
13909:
13897:
13885:
13869:
13857:
13830:
13818:
13806:
13794:
13782:
13757:
13753:
13749:
13730:
13726:Aardal (1995
13724:
13716:
13704:
13677:
13670:Carter (2001
13665:
13638:
13626:
13614:
13590:
13580:
13570:
13564:
13558:
13552:
13550:
13542:
13532:
13529:Varian (1992
13524:
13513:
13501:
13489:
13473:
13437:
13422:Aumann (1966
13402:
13390:
13373:
13354:Bator (1961a
13337:
13324:Bator (
13320:
13299:
13287:
13278:
13269:
13253:
13241:
13229:
13217:
13201:
13161:
13157:
13147:
13122:
13118:
13109:
13090:
13081:
13069:. Retrieved
13064:
13051:
13039:
13027:
13015:
12988:
12976:
12947:
12935:
12920:
12908:
12896:
12884:
12868:
12852:
12836:
12816:
12789:
12782:Carter (2001
12746:
12720:
12670:Starr (2008)
12655:, p. 1)
12639:Starr (1969)
12597:minimization
12590:
12566:
12554:
12542:
12511:
12504:Starr (1969)
12486:
12483:
12469:
12460:
12424:
12414:
12407:
12403:
12396:
12389:
12379:
12372:
12362:
12350:
12343:
12334:
12327:
12321:, then the
12307:
12300:
12296:
12292:
12255:is a finite
12250:
12217:
12213:
12209:
12201:
12186:
12176:
12165:
12161:
12154:dual problem
12135:Ivar Ekeland
12132:
12129:
12118:
12114:
12093:
12089:
12084:
12080:
12066:
12055:
12051:
12044:
12032:
12028:
12025:
11995:
11988:
11981:
11972:
11965:
11961:
11956:
11952:
11919:
11912:
11905:
11901:
11897:
11890:
11886:
11882:
11876:
11858:
11854:
11850:
11842:
11838:
11834:
11828:
11820:
11807:
11803:
11799:
11795:
11791:
11783:
11758:
11754:
11744:
11727:
11723:
11719:
11707:
11698:
11694:
11690:
11682:
11677:
11663:
11642:minimization
11639:
11574:Robert Solow
11572:(1975), and
11556:(2008), and
11554:Paul Krugman
11526:
11522:
11518:
11498:
11488:
11477:
11473:
11471:
11440:
11400:
11388:
11380:
11374:
11372:
11361:
11356:
11344:
11334:
11330:
11328:
11302:
11291:
11286:
11282:
11274:
11267:
11257:
11242:
11235:
11227:
11220:
11213:
11206:
11199:
11165:
11162:Applications
11156:Robert Solow
11154:(1975), and
11138:(2008), and
11136:Paul Krugman
11113:
11105:
11099:
11077:
10950:
10707:
10572:
10309:
10137:, such that
10010:
9929:
9872:
9666:. Now since
9489:
9404:
9125:
9028:, such that
8819:
8816:
8569:
8566:
8496:elements of
8357:
7591:
7437:
7271:
7268:
7056:
7053:
6703:
6450:, represent
6429:
6305:
6285:
6262:
5895:
5782:
5693:
5568:
5334:inner radius
5333:
5331:
5321:
5319:
5266:
5139:
5004:circumradius
5003:
4963:
4875:
4726:
4404:
4154:
4079:
4012:
4004:
3934:
3697:
3504:
3364:
3153:
2883:
2877:
2621:
2554:
2498:
2395:
2344:
2280:
2174:
2046:
1883:
1638:
1395:
1372:
1175:real numbers
1162:
1158:
1150:
1146:
1136:
1130:
1125:
1123:
1093:intersection
1091:
1089:
1080:
1073:
1066:
1058:
1049:
1043:
1036:
1030:
1023:
1017:
1009:
1002:
995:
991:
987:
983:
975:
969:
964:
958:
950:
943:
939:
927:
923:
916:
909:
905:
903:
810:
807:line segment
800:
796:
790:
769:
766:line-segment
761:
741:
738:line segment
733:
689:
682:
675:
604:
598:
594:
590:
586:
582:
575:
569:
567:
558:
551:
544:
537:
530:
523:
516:
509:
499:
495:
491:
488:ordered pair
478:of two
476:vector space
470:
458:
445:
433:
431:
423:
419:real numbers
412:
269:
268:
259:
255:
251:
249:
169:line segment
164:
162:
133:optimization
126:
118:
114:
101:provides an
95:
91:
89:
82:
66:vector space
46:
44:
38:
18:
17605:Duality gap
17600:Dual system
17484:Convex hull
17072:Game theory
17037:Development
16984:Uncertainty
16864:Price floor
16844:Preferences
16783:Competition
16753:Information
16716:Externality
16699:Equilibrium
16640:Transaction
16618:Opportunity
16579:Aggregation
16121:: 220–263.
15136:Howe, Roger
14548:: 100–102.
14516:(1): 1–11.
14358:(1): 1–17.
14324:(1): 1–12.
14101:festschrift
14067:SIAM Review
13890:Ruzsa (1997
13760:: 149–151.
13658:Aubin (2007
13566:Starr (1997
13482:Starr (1969
13443:Wold (1943b
13362:Starr (1969
13258:Wold (1943b
13234:Starr (1969
12969:Starr (1969
12940:Starr (1969
12913:Starr (1969
12877:Starr (1997
12592:quasiconvex
12529:closed sets
12466:John Milton
12431:"bang-bang"
11885:: that is,
11881: f is
11590:game theory
11584:(including
11509:upper bound
11463:Jon Folkman
11364:Herman Wold
11283:preferences
11264:preferences
11100:convexified
11084:Jon Folkman
11072:Jon Folkman
10163:partitions
9279:With that,
8715:, the term
5332:Define the
5002:define its
4642:Similarly,
1938:convex hull
1179:unit circle
1135: Conv(
1132:convex hull
1114:convex hull
1104:Convex hull
713:Convex sets
252:convex hull
153:random sets
111:convex hull
103:upper bound
74:Jon Folkman
31:convex hull
17648:Categories
17528:Radial set
17498:Convex set
17260:Convex set
17102:Managerial
17022:Behavioral
16895:Production
16832:Oligopsony
16672:Elasticity
16584:Budget set
16441:Riemannian
16436:Projective
16421:Symplectic
16416:Hyperbolic
16349:Euclidean
16276:15 January
16249:15 January
16185:15 January
15795:0154.45303
15759:Shubik, M.
15258:, France:
15153:15 January
14948:(5): 493.
14527:2 February
14424:(5): 489.
14188:0313.60012
14018:2 February
13985:References
13914:Vind (1964
13874:Cerf (1999
13862:Weil (1982
12730:scheduling
12694:Lemaréchal
12653:Howe (1979
12525:closed set
12443:continuous
11775:non-convex
11634:convex set
11432:non-convex
11331:non-convex
11323:See also:
11252:See also:
10977:, we have
10363:such that
9747:, we have
9631:such that
8786:is not in
7651:for which
7367:such that
5505:such that
5485:of radius
4405:Note that
4157:to be the
2963:such that
2822:such that
2716:, for any
730:convex set
699:operations
571:multiplied
480:dimensions
17513:Hypograph
17143:Economics
17015:Subfields
16910:Rationing
16827:Oligopoly
16822:Monopsony
16810:Bilateral
16743:Household
16594:Convexity
16462:Lie group
16426:Spherical
15655:Economica
14554:0077-2739
14034:(1980) .
13946:Convexity
13766:0151-0509
13178:0938-2259
13071:1 January
12533:inclusion
12437:, and in
12427:economics
12306:and
12242:principle
12183:non-empty
12166:separable
11883:separable
11761:the graph
11687:arguments
11669:functions
11596:, and in
11467:eponymous
11435:economies
11430:to study
11426:) helped
11347:connected
11260:economics
11182:Economics
11174:, and in
11168:economics
11007:∑
11003:≤
10959:ϵ
10934:ϵ
10891:∑
10887:≤
10879:ϵ
10838:∈
10831:∑
10827:≤
10792:∈
10785:∑
10758:∈
10751:∑
10651:∈
10644:∑
10622:∈
10615:∑
10552:ϵ
10532:ϵ
10321:∈
10285:∈
10239:∈
10220:≤
10110:∈
10103:∑
10081:∈
10074:∑
10024:∈
10018:∀
9962:∑
9958:≤
9898:≤
9858:ϵ
9834:≤
9822:‖
9815:−
9809:‖
9800:≤
9791:−
9715:ϵ
9591:then let
9556:∈
9515:≤
9457:≤
9368:≤
9353:≤
9332:≤
9183:∈
8973:∈
8955:for some
8923:∑
8874:∈
8868:∀
8759:¯
8724:∑
8643:¯
8608:∑
8593:∑
8583:¯
8507:∈
8308:∑
8207:¯
8141:≠
8077:¯
8039:∑
8029:∑
7967:¯
7932:∑
7922:∑
7883:¯
7851:¯
7802:¯
7788:entry of
7711:¯
7610:∈
7504:¯
7451:¯
7394:≥
7340:¯
7305:∑
7295:∑
7285:¯
7243:∈
7225:∈
7199:¯
7166:¯
7152:That is,
7122:¯
7090:∑
7080:∑
7070:¯
6910:¯
6857:¯
6843:. Define
6729:∑
6719:∑
6661:∑
6637:≥
6532:∑
6468:∈
6430:For each
6292:algorithm
6288:existence
6281:Bertsekas
6213:≤
6178:∑
6129:∑
6007:≥
5845:∑
5841:≤
5755:⊂
5714:≤
5626:∖
5591:⊂
5585:⊂
5539:∩
5516:∈
5430:∈
5375:⊂
5276:∑
5208:∑
5201:≤
5124:‖
5118:−
5112:‖
5104:∈
5081:∈
5070:≡
4975:⊂
4627:‖
4621:−
4615:‖
4607:∈
4472:‖
4466:−
4460:‖
4452:∈
4382:‖
4375:−
4369:‖
4356:∈
4344:∈
4268:⊆
4225:‖
4218:−
4212:‖
4204:∈
4126:∈
4091:⊆
3891:∉
3884:∑
3842:∈
3835:∑
3793:…
3775:⊆
3768:⋃
3764:⊆
3729:∑
3660:≤
3654:≤
3615:∈
3582:≤
3576:≤
3523:∈
3457:∑
3423:∑
3128:∈
3088:∖
3055:∈
2972:∑
2901:∈
2831:∑
2780:∈
2727:∈
2657:∑
2513:∑
2297:∈
2253:≤
2247:≤
2221:⊆
2186:∈
2113:∑
2076:∑
1901:⊆
1855:≤
1849:≤
1827:∈
1814:∣
1784:∑
1747:∑
1707:…
1648:∑
1484:∈
1469:∈
1463:∣
1416:⊆
1183:unit disk
915:and
899:vacuously
873:⊘
852:∘
826:∙
795:set
793:non-empty
652:of
494:and
235:⊘
214:∘
188:∙
167:if every
163:A set is
145:functions
129:economics
121:corollary
92:how close
85:dimension
17537:Zonotope
17508:Epigraph
17190:Category
17136:See also
17027:Business
16999:Marginal
16994:Expected
16935:Shortage
16930:Scarcity
16805:Monopoly
16711:Exchange
16623:Implicit
16613:Marginal
16523:Category
16411:Elliptic
16403:geometry
16384:Polyform
16369:Discrete
16351:geometry
16332:Timeline
16316:Geometry
15850:(eds.).
15589:(1997).
15490:(1997).
15352:(eds.).
15336:(1987).
15308:approach
15041:(1974).
14826:(eds.).
13089:(2009).
12587:epigraph
12406:),
12236:, and a
12065:to the "
11867:interval
11746:epigraph
11622:function
11552:(1975),
11548:(1983),
11544:(2005),
11513:distance
11385:Koopmans
11335:separate
11299:function
11294:supports
11232:supports
11158:(1987).
11134:(1975),
11130:(1983),
11126:(2005),
10544:, where
8909:, since
8748:′
8632:′
8551:′
8463:′
8401:′
8339:′
8280:′
8193:′
8063:′
7956:′
7817:, to be
7681:′
7390:′
7329:′
6593:, where
5678:′
5633:′
5581:′
4699:′
4673:′
4363:′
4326:′
4264:′
2890:For any
2277:Notation
2175:for any
1171:interval
1167:integers
963:of
922:in
891:crescent
674:{ (
449:summands
415:interval
278:integers
119:Starr's
107:distance
17689:Sumsets
17592:Duality
17494:Pseudo-
17468:Ursescu
17365:Pseudo-
17339:Concave
17318:Simplex
17298:Duality
17148:Applied
17127:Welfare
16989:Utility
16949:Surplus
16888:Pricing
16800:Duopoly
16793:Perfect
16736:Service
16704:General
16608:Average
16379:Polygon
16327:History
16236:1462618
16155:0064385
16135:0011939
16117:].
16095:0663901
16075:].
16056:2525560
16027:1036734
15993:0737006
15966:1040471
15917:(ed.).
15903:1462618
15831:1909201
15787:1910101
15747:2455326
15712:1216521
15683:0043436
15675:2549499
15619:1475877
15579:1828540
15550:1830308
15510:1451876
15480:0764504
15460:Bibcode
15443:2132405
15326:1113262
15292:0514468
15240:1828536
15196:(ed.).
15182:0700688
15128:1907346
15096:1295240
15061:0389160
15031:1104662
15001:0634800
14962:1828541
14933:1828538
14904:1825163
14846:1727362
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2918:(
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2086:1
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2059:C
2032:.
2029:)
2026:B
2023:(
2019:v
2016:n
2013:o
2010:C
2006:+
2003:)
2000:A
1997:(
1993:v
1990:n
1987:o
1984:C
1980:=
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1974:B
1971:+
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1965:(
1961:v
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1955:o
1952:C
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1904:X
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1895:,
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1864:.
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1840:,
1835:n
1831:Q
1822:n
1818:q
1809:n
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1799:N
1794:1
1791:=
1788:n
1780:{
1777:=
1772:n
1768:Q
1762:N
1757:1
1754:=
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1723:.
1718:N
1714:Q
1710:+
1704:+
1699:2
1695:Q
1691:+
1686:1
1682:Q
1678:=
1673:n
1669:Q
1663:N
1658:1
1655:=
1652:n
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1621:2
1618:,
1615:1
1612:,
1609:0
1606:{
1603:=
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1537:0
1534:{
1531:+
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1519:0
1516:{
1493:.
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1487:B
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1454:x
1451:{
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1442:+
1439:A
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1410:,
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1381:X
1369:.
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1351:]
1347:3
1344:,
1341:1
1338:[
1335:=
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1326:Q
1322:+
1317:1
1313:Q
1290:2
1286:]
1282:2
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1276:1
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1270:=
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1261:Q
1238:2
1234:]
1230:1
1227:,
1224:0
1221:[
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1209:Q
1163:Q
1159:Q
1151:Q
1147:Q
1139:)
1137:Q
1126:Q
1084:D
1081:λ
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1074:λ
1070:0
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1062:d
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999:0
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954:1
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940:λ
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924:Q
920:1
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910:v
906:Q
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785:.
772:.
770:Q
762:Q
744:.
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695:)
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508:(
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375:=
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369:1
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339:0
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324:{
321:=
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315:1
312:,
309:0
306:{
303:+
300:}
297:1
294:,
291:0
288:{
260:Q
256:Q
37:(
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