The Gaussian Minkowski problem for epigraphs of convex functions

2025/08/16 by Li, Xiao, Ye, Deping
#26B25 #35G20 #52A40 #52A41 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2508.12028

Abstract

A variational formula is derived by combining the Gaussian volume of the epigraph of a convex function φ and the perturbation of φ via the infimal convolution. This formula naturally leads to a Borel measure on ℝⁿ and a Borel measure on the unit sphere S^n-1. The resulting Borel measure on ℝⁿ will be called the Euclidean Gaussian moment measure of the convex function φ, 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.

Citations

• The Gauss image problem for pseudo-cones
• The growth rate of surface area measure for noncompact convex sets with prescribed asymptotic cone
• The Riesz α-energy of log-concave functions and related Minkowski problem
• First variation of functional Wulff shapes
• A weighted Minkowski theorem for pseudo-cones
• The dual Orlicz curvature measures for log-concave functions and their related Minkowski problems
• Pseudo-cones
• On the planar Gaussian-Minkowski problem
• Flow by Gauss curvature to the L_p-Gaussian Minkowski problem
• Dual curvature measures for log-concave functions
• Existence of non-symmetric solutions to the Gaussian Minkowski problem
• The anisotropic total variation and surface area measures
• Continuity of the solution to the L_p Minkowski problem in Gaussian probability space
• The Minkowski problem in the Gaussian probability space
• Minkowski type theorems for convex sets in cones
• Geometry of log-concave functions: the L_p Asplund sum and the L_p Minkowski problem
• q-Moment Measures and Applications: A New Approach via Optimal Transport
• General volumes in the Orlicz-Brunn-Minkowski theory and a related Minkowski Problem II
• General volumes in the Orlicz-Brunn-Minkowski theory and a related Minkowski Problem I
• On the general dual Orlicz-Minkowski problem
• A Brunn-Minkowski theory for coconvex sets of finite volume
• The dual Orlicz-Minkowski problem
• Affine hemispheres of elliptic type
• Dealing with moment measures via entropy and optimal transport
• Logarithmically-concave moment measures I
• Moment Measures

Related

• On the planar Gaussian-Minkowski problem
• The Weighted L^p Minkowski Problem
• The Minkowski problem in the Gaussian probability space
• Minkowski Problems for Geometric Measures
• The L_p Gauss dual Minkowski problem