A Minkowski problem for 𝛼-concave functions via optimal transport

Abstract

The notions of the Euclidean surface area measure and the spherical surface area measure of 𝛼-concave functions in ℝₙ, with −1/n < 𝛼 < 0, are introduced via a fi rst 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.