Nguyen Dac Khoi Nguyen

A Minkowski problem for $α$-concave functions via optimal transport

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