The Riesz α-energy of log-concave functions and related Minkowski problem

2024/08/28 by Niufa Fang, Fang, Niufa, Deping Ye +3 · 2 citations
Mathematics · #26B25 #31B99 #35G20 #52A40 #52A41 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Approximation and Integration #Metric Geometry (math.MG) #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2408.16141

openalex publication_date 2024/08/28 · openalex created_date 2024/09/22 · openalex updated_date 2026/07/28

Abstract

We calculate the first order variation of the Riesz α-energy of a log-concave function f with respect to the Asplund sum. Such a variational formula induces the Riesz α-energy measure of log-concave function f, which will be denoted by \mathfrakR_α(f, ⋅). We pose the related Riesz α-energy Minkowski problem aiming to find necessary and/or sufficient conditions on a pregiven Borel measure μ defined on \Rn so that μ=\mathfrakR_α(f,⋅) for some log-concave function f. Assuming enough smoothness, the Riesz α-energy Minkowski problem reduces to a new Monge-Ampère type equation involving the Riesz α-potential. Moreover, this new Minkowski problem can be viewed as a functional counterpart of the recent Minkowski problem for the chord measures in integral geometry posed by Lutwak, Xi, Yang and Zhang (Comm. Pure Appl. Math., 2024). The Riesz α-energy Minkowski problem will be solved under certain mild conditions on μ.

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• The Gaussian Minkowski problem for epigraphs of convex functions
• A Minkowski problem for α-concave functions via optimal transport

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