Minkowski problem

In differential geometry, the Minkowski problem, named after Hermann Minkowski, asks for the construction of a strictly convex compact surface S whose Gaussian curvature is specified. More precisely, the input to the problem is a strictly positive real function ƒ defined on a sphere, and the surface that is to be constructed should have Gaussian curvature ƒ(n(x)) at the point x, where n(x) denotes the normal to S at x. Eugenio Calabi stated: "From the geometric view point it [the Minkowski problem] is the Rosetta Stone, from which several related problems can be solved."

In full generality, the Minkowski problem asks for necessary and sufficient conditions on a non-negative Borel measure on the unit sphere S^n-1 to be the surface area measure of a convex body in ${\displaystyle \mathbb {R} ^{n}}$. Here the surface area measure S_K of a convex body K is the pushforward of the (n-1)-dimensional Hausdorff measure restricted to the boundary of K via the Gauss map. The Minkowski problem was solved by Hermann Minkowski, Aleksandr Danilovich Aleksandrov, Werner Fenchel and Børge Jessen: a Borel measure μ on the unit sphere is the surface area measure of a convex body if and only if μ has centroid at the origin and is not concentrated on a great subsphere. The convex body is then uniquely determined by μ up to translations.

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