The Discrete Gauss Image Problem

Abstract

We study the Discrete Gauss Image Problem, a generalization of Aleksandrov's classical question on the existence of convex bodies with prescribed integral curvature. We introduce a combinatorial problem called the Assignment Problem and show its equivalence to the Discrete Gauss Image Problem. We establish sufficient (and nearly necessary) geometric conditions on measures that solve both problems. Additionally, we provide new discrete interpretations of some classical concepts related to Aleksandrov's integral curtvature, such as, for example, connecting Aleksandrov relation to Hall's Marriage Theorem.