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A GEOMETRIC STUDY OF WASSERSTEIN SPACES: HADAMARD SPACES

JÉRÔME BERTRAND

Institut de Mathématiques, Université Paul Sabatier, 118 Route de Narbonne, F31062 Cedex 9 Toulouse, France

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and

BENOÎT KLOECKNER

Université de Grenoble I, Institut Fourier, CNRS UMR 5582, BP 74, 38402 Saint Martin d'Hères Cedex, France

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https://doi.org/10.1142/S1793525312500227Cited by:19 (Source: Crossref)

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 .

In this paper we investigate the geometry of 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, is not non-positively curved, our results show that have large-scale properties reminiscent of that of X. In particular we define a geodesic boundary for 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 .

Keywords:

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