The Riesz $\alpha$-energy of log-concave functions and related Minkowski problem

TL;DR

This paper introduces a variational formula for the Riesz α-energy of log-concave functions, formulates a Minkowski problem related to this energy, and reduces it to a Monge-Ampère type equation, connecting it to recent integral geometry problems.

Contribution

It establishes a new variational framework for Riesz α-energy of log-concave functions and formulates a Minkowski problem linking it to Monge-Ampère equations and recent geometric measure problems.

Findings

Derived the first order variation formula for Riesz α-energy.

Formulated the Riesz α-energy Minkowski problem for log-concave functions.

Reduced the problem to a Monge-Ampère type equation under smoothness assumptions.

Abstract

We calculate the first order variation of the Riesz $\alpha$-energy of a log-concave function $f$ with respect to the Asplund sum. Such a variational formula induces the Riesz $\alpha$-energy measure of log-concave function $f$, which will be denoted by $\mathfrak{R}_{\alpha}(f, \cdot)$. We pose the related Riesz $\alpha$-energy Minkowski problem aiming to find necessary and/or sufficient conditions on a pregiven Borel measure $\mu$ defined on $\Rn$ so that $\mu=\mathfrak{R}_{\alpha}(f,\cdot)$ for some log-concave function $f$. Assuming enough smoothness, the Riesz $\alpha$-energy Minkowski problem reduces to a new Monge-Amp\`{e}re type equation involving the Riesz $\alpha$-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 $\alpha$-energy Minkowski problem will be solved under certain mild conditions on $\mu$.