Minkowski type theorems for convex sets in cones

TL;DR

This paper extends Minkowski's classical theorems to unbounded convex sets within cones, establishing existence and stability results for their surface area measures in a generalized setting.

Contribution

It introduces new existence and stability theorems for convex sets in cones, broadening Minkowski's classical results beyond bounded convex bodies.

Findings

Established existence conditions for convex sets in cones

Proved stability of surface area measures in this setting

Extended Minkowski's theorems to unbounded convex sets

Abstract

Minkowski's classical existence theorem provides necessary and sufficient conditions for a Borel measure on the unit sphere of Euclidean space to be the surface area measure of a convex body. The solution is unique up to a translation. We deal with corresponding questions for unbounded convex sets, whose behavior at infinity is determined by a given closed convex cone. We provide an existence theorem and a stability result.