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• W2080389441 abstract "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 $${{varphi_{tau} : X to Y : tau > 0}}$$ such that for every $${x,y in X}$$ , $$d_X(x, y) geq tau implies d_Y(varphi_tau (x),varphi_tau (y)) geq |{varphi}_tau|_{rm Lip}tau/K,$$ where $${|{varphi}_{tau}|_{rm Lip}}$$ denotes the Lipschitz constant of $${varphi_{tau}}$$ . We show that if a metric space X threshold-embeds 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 $${X subseteq L_1}$$ threshold-embeds into Hilbert space if and only if X has Markov type 2." @default.
• W2080389441 created "2016-06-24" @default.
• W2080389441 creator A5020836845 @default.
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• W2080389441 date "2013-06-04" @default.
• W2080389441 modified "2023-10-13" @default.
• W2080389441 title "Markov type and threshold embeddings" @default.
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• W2080389441 doi "https://doi.org/10.1007/s00039-013-0234-7" @default.
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