A strategy for non-strictly convex transport costs and the example of ||x-y||p in R2

TL;DR

This paper introduces a decomposition strategy to establish the existence of optimal transport maps for convex but non strictly convex costs, exemplified by ||x-y||^p in R2, with potential applications to convex constrained transport problems.

Contribution

It presents a novel decomposition approach for non-strictly convex costs and demonstrates its effectiveness in proving the existence of optimal transport maps under these conditions.

Findings

Proved existence of optimal transport maps for ||x-y||^p in R2 with p > 1.

Developed a decomposition strategy applicable to convex but non strictly convex costs.

Addressed transport problems with convex displacement constraints.

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 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(||x-y||) with h strictly convex increasing and ||. || an arbitrary norm in \R2.