Calculus of Variations and Geometric Measure Theory

C. Jimenez - F. Santambrogio

Optimal transportation for a quadratic cost with convex constraints and applications

created by santambro on 26 May 2011
modified on 14 Oct 2011


Accepted Paper

Inserted: 26 may 2011
Last Updated: 14 oct 2011

Journal: J. Math. Pures et Appl.
Year: 2011


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 $
^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.

Keywords: optimal transportation, convex bodies, existence of optimal transport maps, c-monotonicity