The Orlicz-Gauss image problem for pseudo-cones and its associated spherical optimal transport

TL;DR

This paper addresses the Orlicz-Gauss image problem for pseudo-cones, establishing existence conditions and linking it to spherical optimal transport using innovative methods that extend previous results.

Contribution

It provides a necessary and sufficient condition for solutions to the Orlicz-Gauss image problem for pseudo-cones and introduces a new approach combining variational and restrictive techniques.

Findings

Established a complete existence criterion for the problem.

Connected the problem to spherical optimal transport.

Extended Schneider's results up to a constant factor.

Abstract

Pseudo-cones serve as the noncompact counterpart of convex bodies in convex geometry. This paper establishes a necessary and sufficient condition for the existence of solutions to the Orlicz-Gauss image problem for pseudo-cones and further demonstrates its connection to spherical optimal transport. Our approach combines the variational method with a novel restrictive technique, thereby strengthening the original result of Schneider up to a constant factor.