An extension of Minkowski's theorem and its applications to questions about projections for measures

Minkowski's Theorem asserts that every centered measure on the sphere which is not concentrated on a great subsphere is the surface area measure of some convex body, and, moreover, the surface area measure determines a convex body uniquely. In this manuscript we prove an extension of Minkowski's theorem. Consider a measure $\mu$ on $\mathbb{R}^n$ with positive degree of concavity and positive degree of homogeneity. We show that a surface area measure of a convex set $K$, weighted with respect to $\mu$, determines a convex body uniquely up to $\mu$-measure zero. We also establish an existence result under natural conditions including symmetry. We apply this result to extend the solution to classical Shephard's problem, which asks the following: if one convex body in $\mathbb{R}^n$ has larger projections than another convex body in every direction, does it mean that the volume of the first convex body is also greater? The answer to this question is affirmative when $n\leq 2$ and negative when $n\geq 3.$ In this paper we introduce a new notion which relates projections of convex bodies to a given measure $\mu$, and is a direct generalization of the Lebesgue area of a projection. Using this notion we state a generalization of the Shephard problem to measures and prove that the answer is affirmative for $n\leq 2$ and negative for $n\geq 3$ for measures which have a positive degree of homogeneity and a positive degree of concavity. We also prove stability and separation results, and establish useful corollaries. Finally, we describe two types of uniqueness results which follow from the extension of Minkowski's theorem.