Continuity of an optimal transport in Monge Problem

Given two absolutely continuous probability measures f ± in R2 , we consider the classical Monge transport problem, with the Euclidean distance as cost function. We prove the existence of a contin- uous optimal transport, under the assumptions that (the densities of) f ± are continuous and strictly positive in the interior part of their supports, and that such supports are convex, compact, and disjoint. We show through several examples that our statement is nearly optimal. Moreover, under the same hypotheses, we also obtain the continuity of the transport density.