Lipschitz continuity of the Wasserstein projections in the convex order on the line

TL;DR

This paper proves that Wasserstein projections in the convex order are Lipschitz continuous in one dimension, providing sharp bounds and advancing understanding in optimal transport and probability measure analysis.

Contribution

It establishes Lipschitz continuity of Wasserstein projections in the convex order on the line, a property not previously known, with sharp bounds demonstrated.

Findings

Wasserstein projections are Lipschitz continuous in dimension one.

Sharp bounds for the 1-Wasserstein distance are provided.

The results contrast with the non-Lipschitz nature of minima and maxima in the convex order.

Abstract

Wasserstein projections in the convex order were first considered in the framework of weak optimal transport, and found application in various problems such as concentration inequalities and martingale optimal transport. In dimension one, it is well-known that the set of probability measures with a given mean is a lattice w.r.t. the convex order. Our main result is that, contrary to the minimum and maximum in the convex order, the Wasserstein projections are Lipschitz continuity w.r.t. the Wasserstein distance in dimension one. Moreover, we provide examples that show sharpness of the obtained bounds for the 1-Wasserstein distance.