A GEOMETRIC STUDY OF WASSERSTEIN SPACES: ULTRAMETRICS by Benôıt

— We study the geometry of the space of measures of a compact ultrametric space X , endowed with the L Wasserstein 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. As a consequence, it is connected by 1 p -Hölder arcs, but any α-Hölder arc with α > 1 p must be constant. This result is obtained via a reformulation of the distance between two measures which is very specific 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} with a natural ultrametric. We are also lead to an example of independent interest: a space of positive lower Minkowski dimension, all of whose proper closed subsets have vanishing lower Minkowski dimension. This work was supported by the Agence Nationale de la Recherche, grant “GMT” ANR-11-JS01-0011. 2 BENOÎT R. KLOECKNER