A geometric study of Wasserstein spaces: ultrametrics

Benoît Kloeckner (1)

1 IF - Institut Fourier

We study the geometry of the space of measures of a compact ultrametric space X, endowed with the L^p Wassertein distance from optimal transportation. We show that the power p of this distance makes this Wasserstein space affinely isometric to a convex subset of l^1. As a consequence, it is connected by 1/p-Hölder arcs, but any a-Hölder arc with a>1/p$ must be constant. This result is obtained via a reformulation of the distance between two measures which is very spectific to the case when X is ultrametric; however thanks to the Mendel-Naor Ultrametric Skeleton it has consequences even when X is a general compact metric space . More precisely, we use it to estimate the size of Wasserstein spaces, measured by an analogue of Hausdorff dimension that is adapted to (some) infinite-dimensional spaces. The result we get generalizes greatly our previous estimate that needed a strong rectifiability assumption. The proof of this estimate involves a structural theorem of independent interest: every ultrametric space contains large co-lipschitz images of regular ultrametric spaces, i.e. spaces of the form {1,...,k}^N with a natural ultrametric.

Mots clés: Optimal transportation; Wasserstein spaces; ultrametric spaces; Hausdorff dimension; critical parameter; ultrametric skeleton; bi-Lipschitz geometry.; bi-Lipschitz geometry

Domaines: Analyse fonctionnelle [math.FA]; Géométrie métrique [math.MG]

Benoît Kloeckner. A geometric study of Wasserstein spaces: ultrametrics. 2013. ⟨hal-00815396v1⟩