L 1 and L ∞ theory

Abstract

Chapter 1 gave, in the framework of the general theory of optimal transportation based on duality methods, an existence result for the optimal transport map when the cost is of the form c(x, y) = | x− y| p, for p∈ ] 1, + ∞[. We look in this chapter at the two limit cases p = 1 and p= ∞, which require additional techniques. We prove existence of an optimal map under absolutely continuous assumptions on the source measure. Then, the discussion section will go into two different directions: on the one hand the L1 and L∞ cases introduced and motivated the study of convex costs which could be non strictly-convex or infinite-valued somewhere; on the other hand one could wonder what is the situation for p < 1, i.e. for costs which are concave increasing functions of the distance.