Partial Regularity for optimal transport maps

TL;DR

This paper proves that optimal transport maps are smooth outside a measure-zero singular set for general costs and Riemannian manifolds, extending regularity results in optimal transport theory.

Contribution

It establishes partial regularity of optimal transport maps for broad classes of cost functions and geometric settings, including Riemannian manifolds.

Findings

Optimal transport maps are smooth outside a measure-zero singular set.

Regularity results extend to general cost functions and Riemannian manifolds.

Singularities are confined to a closed measure-zero set.

Abstract

We prove that, for general cost functions on $\mathbb{R}^n$, or for the cost $d^2/2$ on a Riemannian manifold, optimal transport maps between smooth densities are always smooth outside a closed singular set of measure zero.