The Gauss image problem for pseudo-cones

TL;DR

This paper investigates the Gauss image problem specifically for pseudo-cones, which are unbounded convex sets contained within their recession cones, extending classical convex geometry questions to this broader class.

Contribution

It introduces the Gauss image problem for pseudo-cones, expanding the scope of the classical problem from bounded convex bodies to unbounded pseudo-cones.

Findings

Formulated the Gauss image problem for pseudo-cones

Provided existence conditions for solutions in this setting

Extended convex geometric analysis to unbounded convex sets

Abstract

The Gauss image problem for convex bodies asks for the existence of a convex body that "links" two given measures on the unit sphere in a certain way. We treat here a corresponding question for pseudo-cones, that is, for unbounded closed convex sets strictly contained in their recession cones.