The Gaussian Minkowski Problem for Epigraphs of Convex Functions

Abstract A variational formula is derived by combining the Gaussian volume of the epigraph of a convex function $\varphi $ and the perturbation of $\varphi $ via the infimal convolution. This formula naturally leads to a Borel measure on $\mathbb{R}^{n}$ and a Borel measure on the unit sphere $S^{n-1}$. The resulting Borel measure on $\mathbb{R}^{n}$ will be called the Euclidean Gaussian moment measure of the convex function $\varphi $, and the related Minkowski-type problem will be studied. A solution to the newly posed Minkowski problem will be solved under some mild and natural conditions on the pre-given measure.