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

Abstract

We study p -Wasserstein spaces over the branching spaces and equipped with the maximum norm metric. We show that these spaces are isometrically rigid for all 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 -Wasserstein space over is not rigid. Also, we highlight that the -Wasserstein space is not rigid over the closed interval , while according to our result, its two-dimensional analog, the closed unit ball with the more complicated geodesic structure is rigid.