87:
1833:
1793:
1813:
1803:
1823:
631:
344:
936:
523:
838:
69:
243:
102:(pink oblong), (√3)/2 (grey hexagon), 1/√2 (red octagon) and 1/√3 (orange triangle) calculated from their continued fraction expansions, plotted as slopes
420:
467:
444:
507:
384:
190:
272:
487:
364:
263:
214:
657:). In recent years, Lenstra, Brion, and Barvinok have developed combinatorial theories that enumerate the lattice points in some convex bodies.
626:{\displaystyle \lambda _{1}\lambda _{2}\cdots \lambda _{n}\operatorname {vol} (K)\leq 2^{n}\operatorname {vol} (\mathbb {R} ^{n}/\Gamma ).}
749:
1816:
1516:
993:
946:
1077:
1082:
1420:
1239:
128:
1484:
Hermann Weyl. Theory of reduction for arithmetical equivalence. II . Trans. Amer. Math. Soc. 51 (1942) 203–231.
1234:, London Mathematical Society Lecture Note Series, 89, Cambridge: Cambridge University Press, pp. xii+240,
1547:
1352:
1147:. Springer Classics in Mathematics, Springer-Verlag 1997 (reprint of 1959 and 1971 Springer-Verlag editions).
891:, whose theorem states that the symmetric convex sets that are closed and bounded generate the topology of a
514:
390:
1857:
1806:
1593:
1588:
1573:
1509:
1347:
1342:
1041:(1972), pp. 526–551. See also Schmidt's books; compare Bombieri and Vaaler and also Bombieri and Gubler.
1337:, CBMS-NSF Regional Conference Series in Applied Mathematics 50, SIAM, Philadelphia, Pennsylvania, 1986
42:
32:
219:
1768:
1727:
1606:
144:
1612:
884:
398:
1832:
1555:
1539:
1474:. Theory of reduction for arithmetical equivalence . Trans. Amer. Math. Soc. 48 (1940) 126–164.
1796:
1616:
1565:
1502:
449:
266:
167:
1836:
975:
429:
339:{\displaystyle \operatorname {vol} (K)>2^{n}\operatorname {vol} (\mathbb {R} ^{n}/\Gamma )}
139:
The geometry of numbers has a close relationship with other fields of mathematics, especially
1773:
1702:
1286:
492:
369:
175:
1330:
1278:
1274:
1192:
967:
1763:
1598:
1381:
1315:
1249:
1101:
1003:
676:
121:
1826:
1430:
1373:
8:
1732:
1641:
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1404:
1393:
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1196:
971:
876:
860:
718:
694:
140:
76:
1822:
1188:
1105:
963:
71:
and the study of these lattices provides fundamental information on algebraic numbers.
1778:
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1543:
1443:
1438:
1319:
1150:
1117:
654:
472:
349:
248:
199:
1479:
735:
coefficients and if ε>0 is any given real number, then the non-zero integer points
1692:
1416:
1359:
1235:
1121:
989:
980:, Algorithms and Combinatorics, vol. 2 (2nd ed.), Springer-Verlag, Berlin,
942:
672:
510:
155:
72:
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28:
86:
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20:
883:
in finite-dimensional vector spaces. Minkowski's theorem was generalized to
1753:
1677:
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1471:
1227:
1078:
Computing the continuous discretely: Integer-point enumeration in polyhedra
892:
880:
866:
1265:
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680:
1758:
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1113:
903:
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641:
In 1930–1960 research on the geometry of numbers was conducted by many
922:
1307:
1054:. For more results, see Schneider, and Thompson and see Kalton et al.
1489:
24:
1168:
Cambridge
University Press, New York, 1995. Second edition: 2006.
870:
423:
1494:
833:{\displaystyle |L_{1}(x)\cdots L_{n}(x)|<|x|^{-\varepsilon }}
1657:
1411:. Lecture Notes in Mathematics. Vol. 1467 (2nd ed.).
721:
79:) initiated this line of research at the age of 26 in his work
1024:
Grötschel et al., Lovász et al., Lovász, and Beck and Robins.
269:, sometimes called Minkowski's first theorem, states that if
1128:
875:
Minkowski's geometry of numbers had a profound influence on
636:
517:, is a strengthening of his first theorem and states that
1088:
1050:
For
Kolmogorov's normability theorem, see Walter Rudin's
110:
with errors from their true values (black dashes)
962:
1335:
An
Algorithmic Theory of Numbers, Graphs, and Convexity
879:. Minkowski proved that symmetric convex bodies induce
1672:
1662:
1400:. Lecture Notes in Mathematics 785. Springer. (1980 )
752:
526:
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432:
401:
372:
352:
275:
251:
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178:
45:
1409:
660:
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1201:
Geometric
Algorithms and Combinatorial Optimization
977:
Geometric algorithms and combinatorial optimization
854:
1442:
1279:"Factoring polynomials with rational coefficients"
832:
625:
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461:
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378:
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184:
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898:Researchers continue to study generalizations to
1849:
1262:. Wolters-Noordhoff, North Holland, Wiley. 1969.
1209:Development of the Minkowski Geometry of Numbers
938:Space and Time: Minkowski's papers on relativity
1206:
1092:; Vaaler, J. (Feb 1983). "On Siegel's lemma".
1510:
1230:; Peck, N. Tenney; Roberts, James W. (1984),
122:
1468:Cambridge University Press, Cambridge, 1996.
1461:Cambridge University Press, Cambridge, 1993.
1219:, Johannes SchoiĂźengeier, Rudolf Taschner.
1812:
1802:
1517:
1503:
1459:Convex bodies: the Brunn-Minkowski theory,
1145:An Introduction to the Geometry of Numbers
129:
115:
1358:
934:
637:Later research in the geometry of numbers
599:
315:
225:
48:
1340:
85:
1403:
1183:Handbook of convex geometry. Vol. A. B,
921:MSC classification, 2010, available at
1850:
1223:. Universitext. Springer-Verlag, 1991.
265:is a convex centrally symmetric body.
161:
1498:
1368:, Leipzig and Berlin: R. G. Teubner,
1181:P. M. Gruber, J. M. Wills (editors),
1161:, Springer-Verlag, NY, 3rd ed., 1998.
1221:Geometric and Analytic Number Theory
1159:Sphere Packings, Lattices and Groups
1445:Lectures on the Geometry of Numbers
923:http://www.ams.org/msc/msc2010.html
13:
1083:Undergraduate Texts in Mathematics
614:
496:
373:
330:
179:
14:
1869:
1524:
1480:10.1090/S0002-9947-1940-0002345-2
935:Minkowski, Hermann (2013-08-27).
661:Subspace theorem of W. M. Schmidt
64:{\displaystyle \mathbb {R} ^{n},}
1831:
1821:
1811:
1801:
1792:
1791:
1178:Springer-Verlag, New York, 2007.
855:Influence on functional analysis
689:In the geometry of numbers, the
489:linearly independent vectors of
238:{\displaystyle \mathbb {R} ^{n}}
1213:(Republished in 1964 by Dover.)
1185:North-Holland, Amsterdam, 1993.
1134:Heights in Diophantine Geometry
1069:
90:Best rational approximants for
1570:analytic theory of L-functions
1548:non-abelian class field theory
1057:
1044:
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288:
282:
1:
1176:Convex and discrete geometry,
1075:Matthias Beck, Sinai Robins.
941:. Minkowski Institute Press.
909:
366:contains a nonzero vector in
216:-dimensional Euclidean space
1594:Transcendental number theory
1132:& Walter Gubler (2006).
415:{\displaystyle \lambda _{k}}
7:
1817:List of recreational topics
1589:Computational number theory
1574:probabilistic number theory
1348:Encyclopedia of Mathematics
701:is a positive integer, and
697:in 1972. It states that if
10:
1874:
864:
858:
843:lie in a finite number of
670:
664:
515:Minkowski's second theorem
391:Minkowski's second theorem
388:
165:
33:ring of algebraic integers
1787:
1769:Diophantine approximation
1741:
1728:Chinese remainder theorem
1650:
1532:
1398:Diophantine approximation
986:10.1007/978-3-642-78240-4
885:topological vector spaces
509:. Minkowski's theorem on
462:{\displaystyle \lambda K}
147:, the problem of finding
145:Diophantine approximation
1613:Arithmetic combinatorics
1341:Malyshev, A.V. (2001) ,
1207:Hancock, Harris (1939).
1094:Inventiones Mathematicae
439:{\displaystyle \lambda }
1584:Geometric number theory
1540:Algebraic number theory
925:, Classification 11HXX.
502:{\displaystyle \Gamma }
395:The successive minimum
379:{\displaystyle \Gamma }
185:{\displaystyle \Gamma }
81:The Geometry of Numbers
1703:Transcendental numbers
1617:additive number theory
1566:Analytic number theory
834:
627:
503:
483:
463:
440:
416:
380:
360:
340:
259:
239:
210:
186:
136:
65:
1774:Irrationality measure
1764:Diophantine equations
1607:Hodge–Arakelov theory
1464:Anthony C. Thompson,
1343:"Geometry of numbers"
1287:Mathematische Annalen
1166:Geometric tomography,
1063:Kalton et al. Gardner
1033:Schmidt, Wolfgang M.
1015:Cassels (1971) p. 203
835:
628:
504:
484:
464:
441:
422:is defined to be the
417:
381:
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341:
260:
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211:
187:
89:
73:Hermann Minkowski
66:
1733:Arithmetic functions
1599:Diophantine geometry
1405:Schmidt, Wolfgang M.
1365:Geometrie der Zahlen
1256:C. G. Lekkerkererker
1035:Norm form equations.
972:Schrijver, Alexander
750:
719:linearly independent
677:volume (mathematics)
524:
493:
473:
450:
430:
399:
370:
350:
273:
249:
220:
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176:
43:
1858:Geometry of numbers
1779:Continued fractions
1642:Arithmetic dynamics
1637:Arithmetic topology
1631:P-adic Hodge theory
1623:Arithmetic geometry
1556:Iwasawa–Tate theory
1466:Minkowski geometry,
1439:Siegel, Carl Ludwig
1394:Wolfgang M. Schmidt
1260:Geometry of Numbers
1106:1983InMat..73...11B
1052:Functional Analysis
877:functional analysis
861:normed vector space
695:Wolfgang M. Schmidt
513:, sometimes called
267:Minkowski's theorem
168:Minkowski's theorem
162:Minkowski's results
156:irrational quantity
141:functional analysis
17:Geometry of numbers
1723:Modular arithmetic
1693:Irrational numbers
1627:anabelian geometry
1544:class field theory
1360:Minkowski, Hermann
1300:10.1007/BF01457454
1271:Lenstra, H. W. Jr.
1232:An F-space sampler
1151:John Horton Conway
1114:10.1007/BF01393823
830:
655:Carl Ludwig Siegel
623:
499:
479:
459:
436:
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376:
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182:
137:
61:
1845:
1844:
1742:Advanced concepts
1698:Algebraic numbers
1683:Composite numbers
1136:. Cambridge U. P.
1085:, Springer, 2007.
995:978-3-642-78242-8
964:Grötschel, Martin
961:Schmidt's books.
948:978-0-9879871-1-2
743:coordinates with
511:successive minima
482:{\displaystyle k}
359:{\displaystyle K}
258:{\displaystyle K}
209:{\displaystyle n}
29:algebraic numbers
27:for the study of
1865:
1835:
1825:
1815:
1814:
1805:
1804:
1795:
1794:
1688:Rational numbers
1519:
1512:
1505:
1496:
1495:
1457:Rolf Schneider,
1454:
1448:
1434:
1390:
1389:
1388:
1355:
1327:
1283:
1252:
1228:Kalton, Nigel J.
1212:
1203:, Springer, 1988
1141:J. W. S. Cassels
1137:
1125:
1064:
1061:
1055:
1048:
1042:
1031:
1025:
1022:
1016:
1013:
1007:
1006:
959:
953:
952:
932:
926:
919:
900:star-shaped sets
845:proper subspaces
839:
837:
836:
831:
829:
828:
820:
811:
803:
789:
788:
767:
766:
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693:was obtained by
691:subspace theorem
667:Subspace theorem
651:Harold Davenport
643:number theorists
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151:that approxima
149:rational numbers
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124:
117:
98:(blue diamond),
94:(green circle),
93:
70:
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62:
57:
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1749:Quadratic forms
1737:
1712:P-adic analysis
1668:Natural numbers
1646:
1603:Arakelov theory
1528:
1523:
1490:10.2307/1989946
1451:Springer-Verlag
1423:
1413:Springer-Verlag
1386:
1384:
1281:
1242:
1164:R. J. Gardner,
1155:N. J. A. Sloane
1130:Enrico Bombieri
1090:Enrico Bombieri
1072:
1067:
1062:
1058:
1049:
1045:
1037:Ann. Math. (2)
1032:
1028:
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1019:
1014:
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960:
956:
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904:non-convex sets
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816:
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748:
747:
731:variables with
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431:
428:
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426:of the numbers
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35:is viewed as a
31:. Typically, a
19:is the part of
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11:
5:
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1809:
1807:List of topics
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1708:P-adic numbers
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1552:Iwasawa theory
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1328:
1294:(4): 515–534.
1267:Lenstra, A. K.
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1017:
1008:
994:
968:Lovász, László
954:
947:
927:
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911:
908:
859:Main article:
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685:parallelepiped
673:Siegel's lemma
665:Main article:
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389:Main article:
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166:Main article:
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1754:Modular forms
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1678:Prime numbers
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1560:Kummer theory
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1526:Number theory
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1241:0-521-27585-7
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1217:Edmund Hlawka
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1472:Hermann Weyl
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