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

Abstract

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 the surface area measure determines a convex body uniquely up to a shift. In this manuscript we prove an extension of Minkowski's theorem. Consider a measure μ on Rn 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 μ, determines a convex body uniquely up to μ-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. To do this, we introduce a new notion which relates projections of convex bodies to a given measure μ, and is a generalization of the Lebesgue volume of a projection.