Snowflake Universality of Wasserstein Spaces

For p ∈ (1,∞) let Pp(R) denote the metric space of all p-integrable Borel probability measures on R, equipped with the Wasserstein p metric Wp. We prove that for every ε > 0, every θ ∈ (0, 1/p] and every finite metric space (X, dX), the metric space (X, dX) embeds into Pp(R) with distortion at most 1 + ε. We show that this is sharp when p ∈ (1, 2] in the sense that the exponent 1/p cannot be replaced by any larger number. In fact, for arbitrarily large n ∈ N there exists an n-point metric space (Xn, dn) such that for every α ∈ (1/p, 1] any embedding of the metric space (Xn, d α n) into Pp(R) incurs distortion that is at least a constant multiple of (logn)α−1/p. These statements establish that there exists an Alexandrov space of nonnegative curvature, namely P2(R), with respect to which there does not exist a sequence of bounded degree expander graphs. It also follows that P2(R) 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 P2(R), and the metric cotype dichotomy problem.