A geometric study of Wasserstein spaces: embedding powers

Benoit Kloeckner (1)

1 IF - Institut Fourier

The Wasserstein spaces of a metric space $X$ is a family of sets of sufficiently concentrated probability measures on $X$ endowed with metrics by optimal transport, parametrized by an exponent. In this paper we prove that all powers $X^k$ admit bi-Lipschitz embeddings into the Wasserstein space of $X$, with explicit constants depending only on $k$ and the exponent. As a corollary, we prove that any map acting on a compact space with positive entropy, acts on its measures with positive metric mean dimension (computed with respect to the Wasserstein metric).

Domaines: Géométrie métrique [math.MG]; Systèmes dynamiques [math.DS]; Géométrie différentielle [math.DG]; Analyse fonctionnelle [math.FA]

Benoit Kloeckner. A geometric study of Wasserstein spaces: embedding powers. 2010. ⟨hal-00543790v1⟩