A geometric study of Wasserstein spaces: Hadamard spaces

TL;DR

This paper explores the geometric properties of Wasserstein spaces over Hadamard spaces, revealing large-scale similarities to the base space and establishing non-embeddability results for certain cases.

Contribution

It provides a detailed geometric analysis of Wasserstein spaces over Hadamard spaces, including the construction of a geodesic boundary and non-embeddability theorems.

Findings

W(X) exhibits large-scale properties similar to X

A geodesic boundary for W(X) is constructed

Euclidean plane cannot be isometrically embedded in W(X) if X has the visibility property

Abstract

Optimal transport enables one to construct a metric on the set of (sufficiently small at infinity) probability measures on any (not too wild) metric space X, called its Wasserstein space W(X). In this paper we investigate the geometry of W(X) when X is a Hadamard space, by which we mean that $X$ has globally non-positive sectional curvature and is locally compact. Although it is known that -except in the case of the line- W(X) is not non-positively curved, our results show that W(X) have large-scale properties reminiscent of that of X. In particular we define a geodesic boundary for W(X) that enables us to prove a non-embeddablity result: if X has the visibility property, then the Euclidean plane does not admit any isometric embedding in W(X).