Snowflake universality of Wasserstein spaces

For p ∈ (1; ∞) let P_p(ℝ3) denote the metric space of all p-integrable Borel probability measures on ℝ³, equipped with the Wasserstein p metric W_p. We prove that for every ϵ > 0, every θ ∈ (0; 1/p] and every finite metric space (X, d_X), the metric space (X; d_X ^θ) embeds into P_p(ℝ³) 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 (X_n, d_n) such that for every α ∈ (1/p, 1] any embedding of the metric space (X_n, d_n ^α) into P_p (ℝ³) incurs distortion that is at least a constant multiple of (log n)^α-1/p. These statements establish that there exists an Alexandrov space of nonnegative curvature, namely P₂(ℝ³), with respect to which there does not exist a sequence of bounded degree expander graphs. It also follows that P₂(ℝ³) 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^k), and the metric cotype dichotomy problem.