Existence of convex hypersurfaces with prescribed centroaffine curvature

Abstract

In this paper, we study the existence of solutions to the centroaffine Minkowski problem, namely the existence of closed convex hypersurfaces in the Euclidean space R n + 1 \mathbb {R}^{n+1} with prescribed centroaffine curvature. This problem can be described as a variational problem with a functional resembling Kantorovich’s dual functional in optimal transportation. By using the Gauss curvature flow, we obtain new conditions for the existence of solutions to the problem.