A proof of Sudakov theorem with strictly convex norms

Abstract

The paper establishes a solution to the Monge problem in \({\mathbb {R}^n}\) for a possibly asymmetric norm cost function and absolutely continuous initial measures, under the assumption that the unit ball is strictly convex—but not necessarily differentiable nor uniformly convex. The proof follows the strategy initially proposed by Sudakov in 1976, found to be incomplete in 2000; the missing step is fixed in the above case adapting a disintegration technique introduced for a variational problem. By strict convexity, mass moves along rays, and we also investigate the divergence of the vector field of rays.