The Monge problem in Rd

Abstract

We first consider the Monge problem in a convex bounded subset of ${\mathbb{R}}^d$. The cost is given by a general norm, and we prove the existence of an optimal transport map under the classical assumption that the first marginal is absolutely continuous with respect to the Lebesgue measure. In the final part of the paper we show how to extend this existence result to a general open subset of ${\mathbb{R}}^d$.