Optimal transportation for a quadratic cost with convex constraints and applications

Abstract

We 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 $+\infty$ otherwise. The result is proven for $C$ satisfying some technical assumptions allowing any convex body in ${\mathbb{R}}^2$ and any convex polyhedron in ${\mathbb{R}}^d$, $d>2$. 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 $L^{\infty }$ problems and, later on, with the Monge problem for arbitrary norms.