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.