Geometric and Functional Analysis

, Volume 18, Issue 5, pp 1609–1659 | Cite as

Trees and Markov Convexity

  • James R. Lee
  • Assaf NaorEmail author
  • Yuval Peres


We show that an infinite weighted tree admits a bi-Lipschitz embedding into Hilbert space if and only if it does not contain arbitrarily large complete binary trees with uniformly bounded distortion. We also introduce a new metric invariant called Markov convexity, and show how it can be used to compute the Euclidean distortion of any metric tree up to universal factors.

AMS Mathematics Subject Classification:

51F99 05C12 05C05 60B99 

Keywords and phrases:

Metric trees bi-Lipschitz embeddings uniform convexity 


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Copyright information

© Birkhaeuser 2008

Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringUniversity of WashingtonSeattleUSA
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  3. 3.Microsoft ResearchRedmondUSA

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