A strategy for non-strictly convex transport costs and the example of $║x-y║^p$ in $R^2$

Abstract

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 a consequence of our procedure, we have to treat some transport problems, of independent interest, with a convex constraint on the displacement. To illustrate possible results obtained through this general approach, 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 $h║x-y║$, with h strictly convex increasing and $║.║$ an arbitrary norm in $R^2$.