Markov Type And Threshold Embeddings

For two metric spaces X and Y , say that X threshold-embeds into Y if there exist a number K > 0 and a family of Lipschitz maps {ϕτ : X → Y : τ > 0} such that for every x, y ∈ X, dX(x, y) ≥ τ =⇒ dY (ϕτ (x), ϕτ (y)) ≥ kϕτ kLipτ /K , where kϕτ kLip denotes the Lipschitz constant of ϕτ . We show that if a metric space X thresholdembeds into a Hilbert space, then X has Markov type 2. As a consequence, planar graph metrics and doubling metrics have Markov type 2, answering questions of Naor, Peres, Schramm, and Sheffield. More generally, if a metric space X threshold-embeds into a p-uniformly smooth Banach space, then X has Markov type p. Our results suggest some non-linear analogs of Kwapien’s theorem. For instance, a subset X ⊆ L1 threshold-embeds into Hilbert space if and only if X has Markov type 2.