Optimal transportation for a quadratic cost with convex constraints and applications

AbstractWe prove existence of an optimal transport map in the Monge–Kantorovich 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 Rd, 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