A Minkowski problem for $\alpha$-concave functions via 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.