The Gauss Image Problem with Weak Aleksandrov Condition

TL;DR

This paper introduces a weaker condition for the Gauss Image Problem, establishing its necessity and providing solutions when one measure is discrete and the other is absolutely continuous, expanding the problem's solvability scope.

Contribution

It proposes a relaxed Aleksandrov condition for the Gauss Image Problem and demonstrates its necessity and sufficiency in specific measure cases.

Findings

The weaker condition is necessary for measure relations via convex bodies.

Solutions are obtained when one measure is discrete and the other is absolutely continuous.

The new condition broadens the class of measure pairs for which the problem can be solved.

Abstract

We introduce a relaxation of the Aleksandrov condition for the Gauss Image Problem. This weaker condition turns out to be a necessary condition for two measures to be related by a convex body. We provide several properties of the new condition. A solution to the Gauss Image Problem is obtained for the case when one of the measures is assumed to be discrete and the another measure is assumed to be absolutely continuous, under the new relaxed assumption.