The Gaussian Minkowski problem for epigraphs of convex functions

Xiao Li; Deping Ye

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arXiv · 2508.12028

The Gaussian Minkowski problem for epigraphs of convex functions

Xiao Li·Deping Ye

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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.

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BibTeX RIS

Xiao Li, Deping Ye. 2025-08-16. The Gaussian Minkowski problem for epigraphs of convex functions. https://arxiv.org/abs/2508.12028

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