A Minkowski problem for $\hat{I}\pm$-concave functions via optimal transport

Abstract

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