Optimal transportation for a quadratic cost with convex constraints and applications

We prove existence of an optimal transport map in the MongeKantorovich problem associated to a cost c(x, y) which is not finite everywhere, but coincides with |x− y|2 if the displacement y − x belongs to a given convex set C and it is +∞ otherwise. The result is proven for C satisfying some technical assumptions allowing any convex body in R2 and any convex polyhedron in R, d > 2. The tools are inspired by the recent Champion-DePascale-Juutinen technique. Their idea, based on density points and avoiding disintegrations and dual formulations, allowed to deal with L∞ problems and, later on, with the Monge problem for arbitrary norms.