On the Gauss image problem

TL;DR

This paper addresses the Gauss image problem by providing a solution when given two Borel measures on the sphere, with one measure absolutely continuous relative to the uniform measure, advancing understanding in convex geometry.

Contribution

It offers a novel solution to the Gauss image problem under specific measure conditions, expanding the scope of solvable cases in convex geometric analysis.

Findings

Solved the Gauss image problem for measures with one absolutely continuous measure

Extended the class of measures for which the problem can be solved

Provided new insights into the measure-theoretic aspects of convex geometry

Abstract

In this note, we solve the Gauss image problem given two Borel measures on the unit sphere, one of which is absolutely continuous with respect to the uniform measure.