Existence of solutions to the Gaussian dual Minkowski problem

Abstract

Gaussian dual curvature measure is introduced and Gaussian dual Minkowski problem is studied. This problem amounts to solving a class of Monge-Ampère type equations on the unit sphere. Existence and uniqueness of solutions to the relevant Monge-Ampère type equations are obtained in the smooth category when $q\le 0$, respectively. For $q<0$, a complete solution to existence part of the Gaussian dual Minkowski problem is presented. For the case of $q=0$, a weak solution to the Monge-Ampère type equation related to this problem is provided when given measure has the density f which is sandwiched between two positive constants belonging to the interval 0 to 1.