A Minkowski problem for $\alpha$-concave functions via optimal transport

TL;DR

This paper introduces surface area measures for lpha-concave functions and measures, extending classical concepts, and solves the Minkowski problem for these measures using optimal transport techniques.

Contribution

It defines new surface area measures for lpha-concave functions and measures and addresses the Minkowski problem for these measures through optimal transport methods.

Findings

Defined Euclidean and spherical surface area measures for lpha-concave functions.

Extended surface area measures to lpha-concave measures.

Solved the Minkowski problem for lpha-concave measures using optimal transport.

Abstract

The notions of the Euclidean surface area measure and the spherical surface area measure of $\alpha$-concave functions in $\mathbb{R}^n$, with $-\frac{1}{n}<\alpha<0$, are introduced via a first variation of the total mass functional with respect to the $\alpha$-sum operation. Subsequently, these notions are extended to those for $\alpha$-concave measures. We then study the Minkowski problem associated with the Euclidean surface area measures of $\alpha$-concave measures via optimal transport.