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

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-Ampère 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 $\mu$.