Existence of solutions to the even Gaussian dual Minkowski problem

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 R-n such that its Gaussian dual curvature measure C-<combining double overline>gamma n,C-q (K, <middle dot>) 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. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.