Optimal transportation for a quadratic cost with convex constraints and applications

TL;DR

This paper establishes the existence of an optimal transport map for a quadratic cost constrained by convex sets, extending classical results to cases with non-finite costs outside these sets.

Contribution

It proves existence results for optimal transport maps under convex constraints on the displacement, broadening applicability to convex bodies and polyhedra in various dimensions.

Findings

Existence of optimal transport maps under convex set constraints.

Extension of classical quadratic cost problems to non-finite cost scenarios.

Application of Champion-DePascale-Juutinen technique to constrained optimal transport.

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 $\R^2$ and any convex polyhedron in $\R^d$, $d>2$. 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.