On the degeneracy of optimal transportation

TL;DR

This paper extends a known upper bound on the degeneration of optimal transport maps from quadratic costs to more general costs satisfying certain curvature conditions, broadening the understanding of transport map regularity.

Contribution

It generalizes Caffarelli's dimensional upper bound to a wider class of cost functions meeting Ma, Trudinger, and Wang's curvature condition.

Findings

Extended the upper bound to costs satisfying the curvature condition

Broadened the applicability of regularity results in optimal transport

Provided theoretical insights into transport map degeneracy under general costs

Abstract

We extend a dimensional upper bound on how much an optimal transport map can degenerate for the quadratic transportation cost, originally due to Caffarelli, to cost functions that satisfy the curvature condition of Ma, Trudinger, and Wang.