Existence of solutions to the even Gaussian dual Minkowski problem

Abstract

In this paper, we consider the Gaussian dual Minkowski problem. The problem involves a new type of fully nonlinear partial differential equations on the unit sphere. Our main purpose is to show the existence of solutions to the even Gaussian dual Minkowski problem for q>0. More precisely, we will show that there exists an origin-symmetric convex body K in Rn such that its Gaussian dual curvature measure C˜γn,q(K,⋅) has density f (up to a constant) on the unit sphere when q>0 and f has positive upper and lower bounds. Note that if f is smooth then K is also smooth. As the application of smooth solutions, we completely solve the even Gaussian dual Minkowski problem for q>0 based on an approximation argument.