A geometric study of Wasserstein spaces: Hadamard spaces

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).