Trees and Markov convexity

James R. Lee; Assaf Naor; Yuval Peres

doi:10.1007/s00039-008-0689-0

Trees and Markov convexity

James R. Lee, Assaf Naor, Yuval Peres

Mathematics

Research output: Contribution to journal › Article › peer-review

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.

Original language	English (US)
Pages (from-to)	1609-1659
Number of pages	51
Journal	Geometric and Functional Analysis
Volume	18
Issue number	5
DOIs	https://doi.org/10.1007/s00039-008-0689-0
State	Published - Feb 2009

Keywords

Metric trees
Uniform convexity
bi-Lipschitz embeddings

ASJC Scopus subject areas

Analysis
Geometry and Topology

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10.1007/s00039-008-0689-0

Cite this

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title = "Trees and Markov convexity",

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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AB - 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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KW - Uniform convexity

KW - bi-Lipschitz embeddings

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Trees and Markov convexity

Abstract

Keywords

ASJC Scopus subject areas

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