The Gaussian Minkowski problem for epigraphs of convex functions

TL;DR

This paper introduces a Minkowski-type problem for convex functions' epigraphs, deriving a variational formula that links Gaussian volume with measures, and solves the problem under natural conditions.

Contribution

It formulates and solves a new Minkowski problem for Euclidean Gaussian moment measures associated with convex functions.

Findings

Derived a variational formula connecting Gaussian volume and perturbations of convex functions.

Established existence results for the Minkowski problem under mild conditions.

Introduced the concept of Euclidean Gaussian moment measure for 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. In particular, the newly posed Minkowski problem is solved under some mild and natural conditions on the pre-given measure.