A geometric study of Wasserstein spaces: Euclidean spaces

Benoit Kloeckner (1)

1 IF - Institut Fourier

We study the Wasserstein space (with quadratic cost) of Euclidean spaces as an intrinsic metric space. In particular we compute their isometry groups. Surprisingly, in the case of the line, there exists a (unique) ``exotic'' isometric flow. This contrasts with the case of higher-dimensional Euclidean spaces, where all isometries of the Wasserstein space preserve the shape of measures. We also study the curvature and various ranks of these spaces.

Mots clés: Wasserstein metric; Optimal transportation; Isometry group; rank

Domaines: Géométrie métrique [math.MG]

Benoit Kloeckner. A geometric study of Wasserstein spaces: Euclidean spaces. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 2010, IX (2), pp.297-323. ⟨10.2422/2036-2145.2010.2.03⟩. ⟨hal-00275067v3⟩