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published the solutions of two long standing open problems, the Weyl problem and the
Minkowski problem in Euclidean 3-space. L. Nirenberg's solution of the Minkowski problem was a milestone in global geometry. He has been selected to be the first recipient of the Chern Medal (in 2010) for his role in
198:: restoration of convex shape over the given Gauss surface curvature. The inverse problem of the short-wave diffraction is reduced to the Minkowski problem. The Minkowski problem is the basis of the mathematical theory of
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the formulation of the modern theory of non-linear elliptic partial differential equations, particularly for solving the Weyl problem and the
Minkowski problems in Euclidean 3-space.
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The
Minkowski problem, despite its clear geometric origin, is found to have its appearance in many places. The problem of
220:(1973) for resolving the multidimensional Minkowski problem in Euclidean spaces. Pogorelov resolved the Weyl problem in
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has centroid at the origin and is not concentrated on a great subsphere. The convex body is then uniquely determined by
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Cheng, Shiu Yuen; Yau, Shing Tung (1976). "On the regularity of the solution of the n-dimensional
Minkowski problem".
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gives a complete proof of the higher-dimensional
Minkowski problem in Euclidean spaces. Shing-Tung Yau received the
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asks for necessary and sufficient conditions on a non-negative Borel measure on the unit sphere
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Nirenberg, L. (1953). "The Weyl and
Minkowski problems in differential geometry in the large".
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is specified. More precisely, the input to the problem is a strictly positive real function
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Constructing a strictly convex compact surface with specified
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on the unit sphere is the surface area measure of a convex body if and only if
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defined on a sphere, and the surface that is to be constructed should have
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Minkowski’s and related problems for convex surfaces with boundaries
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151:-dimensional Hausdorff measure restricted to the boundary of
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202:as well as for the physical theory of diffraction.
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88:stated: "From the geometric view point it is the
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194:is easily reduced to the Minkowski problem in
326:Bulletin of the American Mathematical Society
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364:Convex Bodies: the Brunn-Minkowski Theory
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240:International Congress of Mathematicians
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248:elliptic partial differential equations
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413:The Minkowsky Multidimensional Problem
323:, by Aleksey Vasil'yevich Pogorelov",
321:The Minkowski multidimensional problem
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159:. The Minkowski problem was solved by
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258:in Euclidean spaces concerning the
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242:in Warsaw in 1982 for his work in
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165:Aleksandr Danilovich Aleksandrov
136:. Here the surface area measure
129:{\displaystyle \mathbb {R} ^{n}}
340:10.1090/S0273-0979-1979-14645-7
514:Partial differential equations
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478:Michigan Mathematical Journal
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411:Pogorelov, A. V. (1979)
362:Schneider, Rolf (1993),
285:"Volumen und Oberfläche"
95:In full generality, the
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415:, Washington: Scripta,
453:10.1002/cpa.3160290504
441:Comm. Pure Appl. Math.
391:10.1002/cpa.3160060303
283:Minkowski, H. (1903).
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509:Differential geometry
379:Comm. Pure Appl. Math
290:Mathematische Annalen
264:Monge–Ampère equation
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21:differential geometry
504:Theorems in geometry
187:up to translations.
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319:(1979), "Review of
230:'s joint work with
218:Ukraine State Prize
303:10.1007/BF01445180
175:: a Borel measure
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51:Gaussian curvature
43:Gaussian curvature
519:Hermann Minkowski
260:Dirichlet problem
256:Hermann Minkowski
252:Calabi conjecture
196:Euclidean 3-space
161:Hermann Minkowski
143:of a convex body
97:Minkowski problem
29:Hermann Minkowski
25:Minkowski problem
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333:: 636–639,
200:diffraction
105:convex body
498:Categories
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270:References
224:in 1969.
216:received
157:Gauss map
205:In 1953
155:via the
80:at
68:, where
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472:(1959)
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238:at the
36:surface
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