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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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