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Trees and Markov Convexity

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Abstract.

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.

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Authors and Affiliations

  1. Department of Computer Science and Engineering, University of Washington, Box 352350, Seattle, WA, 98195-2350, USA

    James R. Lee

  2. Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY, 10012, USA

    Assaf Naor

  3. Microsoft Research, One Microsoft Way, Redmond, WA, 98052-6399, USA

    Yuval Peres

Authors
  1. James R. Lee
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  2. Assaf Naor
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  3. Yuval Peres
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Corresponding author

Correspondence to Assaf Naor.

Additional information

Received: September 2006, Revision: January 2007, Accepted: May 2007

The research of the first author was conducted while he was at U. C. Berkeley and the Institute for Advanced Study.

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Lee, J.R., Naor, A. & Peres, Y. Trees and Markov Convexity. GAFA Geom. funct. anal. 18, 1609–1659 (2009). https://doi.org/10.1007/s00039-008-0689-0

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  • DOI: https://doi.org/10.1007/s00039-008-0689-0

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