This is an announcement for the paper "Snowflake universality of Wasserstein spaces" by Alexandr Andoni, Assaf Naor, and Ofer Neiman.

Abstract: For $p\in (1,\infty)$ let $\mathscr{P}_p(\mathbb{R}^3)$ denote the metric space of all $p$-integrable Borel probability measures on $\mathbb{R}^3$, equipped with the Wasserstein $p$ metric $\mathsf{W}_p$. We prove that for every $\varepsilon>0$, every $\theta\in (0,1/p]$ and every finite metric space $(X,d_X)$, the metric space $(X,d_{X}^{\theta})$ embeds into $\mathscr{P}_p(\mathbb{R}^3)$ with distortion at most $1+\varepsilon$. We show that this is sharp when $p\in (1,2]$ in the sense that the exponent $1/p$ cannot be replaced by any larger number. In fact, for arbitrarily large $n\in \mathbb{N}$ there exists an $n$-point metric space $(X_n,d_n)$ such that for every $\alpha\in (1/p,1]$ any embedding of the metric space $(X_n,d_n^\alpha)$ into $\mathscr{P}_p(\mathbb{R}^3)$ incurs distortion that is at least a constant multiple of $(\log n)^{\alpha-1/p}$. These statements establish that there exists an Alexandrov space of nonnegative curvature, namely $\mathscr{P}_{! 2}(\mathbb{R}^3)$, with respect to which there does not exist a sequence of bounded degree expander graphs. It also follows that $\mathscr{P}_{! 2}(\mathbb{R}^3)$ does not admit a uniform, coarse, or quasisymmetric embedding into any Banach space of nontrivial type. Links to several longstanding open questions in metric geometry are discussed, including the characterization of subsets of Alexandrov spaces, existence of expanders, the universality problem for $\mathscr{P}_{! 2}(\mathbb{R}^k)$, and the metric cotype dichotomy problem.

Archive classification: math.MG math.FA

Submitted from: naor@math.princeton.edu

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1509.08677

or

http://arXiv.org/abs/1509.08677