[1] 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.
The Minkowski problem was solved by Hermann Minkowski, Aleksandr Danilovich Aleksandrov, Werner Fenchel and Børge Jessen:[3] 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 Minkowski problem, despite its clear geometric origin, is found to have its appearance in many places.
[4] A. V. Pogorelov received Ukraine State Prize (1973) for resolving the multidimensional Minkowski problem in Euclidean spaces.
[5] Shing-Tung Yau's joint work with Shiu-Yuen Cheng gives a complete proof of the higher-dimensional Minkowski problem in Euclidean spaces.