Regularity for a log-concave to log-concave mass transfer problem with near Euclidean cost

TL;DR

This paper demonstrates that optimal transport maps between Gaussian measures remain regular under costs close to Euclidean and in local neighborhoods with smooth costs, extending regularity results in optimal transport theory.

Contribution

It establishes regularity of optimal maps for near Euclidean costs and local smooth costs on manifolds, broadening the understanding of regularity conditions in optimal transport.

Findings

Optimal maps are regular when cost is close to Euclidean.

Gaussian measures on manifolds can be transported smoothly in local neighborhoods.

Regularity persists under small perturbations of the cost function.

Abstract

If the cost function is not too far from the Euclidean cost, then the optimal map transporting Gaussians restricted to a ball will be regular. \ Similarly, given any cost function which is smooth in a neighborhood of two points on a manifold, there are small neighborhoods near each such that a Gaussian restricted to one is transported smoothly to a Gaussian on the other