The Gauss Image Problem with Weak Aleksandrov Condition

Abstract

We introduce a relaxation to the Aleksandrov relation assumption for the Gauss Image Problem. This new assumption turns out to be a necessary assumption 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 of when one of the measures is assumed to be discrete and another measure is assumed to be absolutely continuous, under the new relaxed assumption.