Minkowski problem

Remarks

The problem is solved. A convex body $K$ with $S_K = \mu$ exists if and only if $\mu$ is not concentrated on a great subsphere of $S^{n-1}$ and $\int_{S^{n-1}} u \, d\mu(u) = 0$. The body is then unique up to translation. Minkowski solved the case of polytopes. Let $u_1, \ldots, u_m$ be distinct unit vectors spanning $\R^n$, and let $a_1, \ldots, a_m > 0$ with $\sum_i a_i u_i = 0$. Then there is a polytope, unique up to translation, whose facets have outer normals $u_i$ and areas $a_i$. The general case is due to Minkowski, Alexandrov, and Fenchel and Jessen. When $\mu$ has a positive density $f$, the problem asks for a body whose Gaussian curvature at the point with outer normal $u$ is $1 / f(u)$. This is a Monge–Ampère equation on the sphere. The regularity of its solutions was studied by Pogorelov and by Cheng and Yau. Example: the uniform measure $\mu = c \, \sigma$ on $S^{n-1}$, with $\sigma$ the surface measure and $c > 0$, is the surface area measure of a ball of radius $c^{1/(n-1)}$.