Isometric rigidity of the Wasserstein space over the plane with the maximum metric

TL;DR

This paper proves that Wasserstein spaces over the plane with the maximum metric are isometrically rigid for all p ≥ 1, meaning all their isometries come from isometries of the underlying space, contrasting with the Euclidean case.

Contribution

It establishes the isometric rigidity of Wasserstein spaces over branching spaces with the maximum metric for all p ≥ 1, a novel result in metric geometry.

Findings

Wasserstein spaces over the maximum metric are isometrically rigid for all p ≥ 1.

This rigidity contrasts with the Euclidean case where the 2-Wasserstein space is not rigid.

The 1-Wasserstein space over the interval is not rigid, but the 2D analog over the square is rigid.

Abstract

We study $p$-Wasserstein spaces over the branching spaces $\mathbb{R}^2$ and $[-1,1]^2$ equipped with the maximum norm metric. We show that these spaces are isometrically rigid for all $p\geq1,$ meaning that all isometries of these spaces are induced by isometries of the underlying space via the push-forward operation. This is in contrast to the case of the Euclidean metric since with that distance the $2$-Wasserstein space over $\mathbb{R}^2$ is not rigid. Also, we highlight that the $1$-Wasserstein space is not rigid over the closed interval $[-1,1]$, while according to our result, its two-dimensional analog, the closed unit ball $[-1,1]^2$ with the more complicated geodesic structure is rigid.