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532:{\displaystyle \mathrm {vol} {\big (}(1-\lambda )K+\lambda L{\big )}^{1/n}\geq (1-\lambda )\mathrm {vol} (K)^{1/n}+\lambda \,\mathrm {vol} (L)^{1/n},}
554:
allowed connections to be made between geometry and analysis, to the extent that in an integral form of the Brunn–Minkowski inequality known as the
665:, series Die Grundlehren der mathematischen Wissenschaften, vol. Band 153, New York: Springer-Verlag New York Inc., pp. xiv+676,
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in statistics. The proof of the Brunn–Minkowski inequality predates modern measure theory; the development of measure theory and
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Characterization of rectifiability through existence of approximate tangents, densities, projections, etc.
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for the kind of singularities that can occur in these more general soap films and soap bubbles clusters.
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without topological restrictions, thus sparking geometric measure theory. Later
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Principles in Topology (Multidimensional Minimal Surface Theory)
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Federer, H. (1978), "Colloquium lectures on geometric measure theory",
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with which they were able to solve the orientable
Plateau's problem
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with the least possible regularity required to admit approximate
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can be proved on a single page and quickly yields the classical
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The following objects are central in geometric measure theory:
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Gardner, Richard J. (2002), "The Brunn-Minkowski inequality",
603:; Fleming, Wendell H. (1960), "Normal and integral currents",
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Plateau type minimization problems from calculus of variations
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Flat chains, an alternative generalization of the concept of
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Geometric measure theory was born out of the desire to solve
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The problem had remained open since it was posed in 1760 by
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Study of geometric properties of sets through measure theory
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Rectifiability and uniform rectifiability of (subsets of)
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The following theorems and concepts are also central:
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773:Geometry of Sets and Measures in Euclidean Spaces
75:) which asks if for every smooth closed curve in
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47:. It allows mathematicians to extend tools from
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127:. It was solved independently in the 1930s by
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794:Geometric measure theory: A beginner's guide
116:equals the given curve. Such surfaces mimic
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865:Encyclopedia of Mathematics
738:(3): 355–405 (electronic),
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556:Prékopa–Leindler inequality
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860:"Geometric measure theory"
337:Brunn–Minkowski inequality
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51:to a much larger class of
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55:that are not necessarily
663:Geometric measure theory
544:isoperimetric inequality
343:-dimensional volumes of
306:isoperimetric inequality
209:Orthogonal projections,
25:geometric measure theory
883:Peter Mörters' GMT page
858:O'Neil, T.C. (2001) ,
689:Bull. Amer. Math. Soc.
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219:Uniform rectifiability
139:restrictions. In 1960
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646:. The first paper of
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49:differential geometry
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184:Hausdorff dimension
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31:) is the study of
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153:analytically
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211:Kakeya sets
189:Rectifiable
157:Jean Taylor
137:topological
21:mathematics
897:Categories
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594:References
133:Tibor RadĂł
118:soap films
43:) through
870:EMS Press
754:0273-0979
488:λ
444:λ
441:−
432:≥
404:λ
392:λ
389:−
265:manifolds
251:manifolds
240:manifolds
191:sets (or
108:of least
33:geometric
908:Geometry
792:(2009),
771:(1999),
712:(1990),
661:(1969),
652:currents
578:Currents
562:See also
339:for the
331:Examples
271:applies.
255:boundary
244:boundary
237:oriented
233:Currents
149:currents
125:Lagrange
114:boundary
53:surfaces
851:0428181
843:1970949
812:2455580
762:1898210
681:0257325
648:Federer
635:0123260
627:1970227
163:proved
106:surface
63:History
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667:ISBN
350:and
335:The
314:area
304:The
293:The
282:The
197:sets
182:and
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131:and
110:area
37:sets
831:doi
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639:Zbl
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29:GMT
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