The Discrete Gauss Image Problem

TL;DR

This paper investigates the Discrete Gauss Image Problem, linking it to a combinatorial assignment problem, and establishes geometric conditions for solutions, offering new interpretations of classical curvature concepts.

Contribution

It introduces the Assignment Problem and proves its equivalence to the Discrete Gauss Image Problem, providing geometric conditions and new discrete interpretations of classical curvature concepts.

Findings

Established equivalence between the Assignment Problem and the Discrete Gauss Image Problem.

Derived sufficient and nearly necessary geometric conditions for solutions.

Connected Aleksandrov's relation to Hall's Marriage Theorem.

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.