Calculus of Variations and Geometric Measure Theory

G. Carlier - L. De Pascale - F. Santambrogio

A strategy for non-strictly convex transport costs and the example of $||x-y||^p$ in ${\mathbb R}^2$

created by depascal on 31 Aug 2009
modified on 28 Jan 2010


Accepted Paper

Inserted: 31 aug 2009
Last Updated: 28 jan 2010

Journal: Communications in Mathematical Sciences
Pages: 9
Year: 2009


This paper deals with the existence of optimal transport maps for some optimal transport problems with a convex but non strictly convex cost. We give a decomposition strategy to address this issue. As part of our strategy, we have to treat some transport problems, of independent interest, with a convex constraint on the displacement. As an illustration of our strategy, we prove existence of optimal transport maps in the case where the source measure is absolutely continuous with respect to the Lebesgue measure and the transportation cost is of the form $h(\Vert x-y\Vert)$ with $h$ strictly convex increasing and $\Vert . \Vert$ an arbitrary norm in $\R^2$.

Keywords: Monge problem, Monge-Kantorovich problem, optimal transport problem