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A geometric study of Wasserstein spaces: Euclidean spaces. (English) Zbl 1218.53079
Summary: We consider Wasserstein spaces (with quadratic transportation cost) as intrinsic metric spaces. We are interested in usual geometric properties: curvature, rank and isometry group, mostly in the case of Euclidean spaces. Our most striking result is that the Wasserstein space of the line admits “exotic” isometries, which do not preserve the shape of measures.

MSC:
53C70 Direct methods (\(G\)-spaces of Busemann, etc.)
54E70 Probabilistic metric spaces
28A33 Spaces of measures, convergence of measures
60B05 Probability measures on topological spaces
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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