Scale-oblivious metric fragmentation and the nonlinear Dvoretzky theorem

Assaf Naor, Terence Tao

Research output: Contribution to journalArticle

Abstract

We introduce a randomized iterative fragmentation procedure for finite metric spaces, which is guaranteed to result in a polynomially large subset that is D-equivalent to an ultrametric, where D ∈ (2,∞) is a prescribed target distortion. Since this procedure works for D arbitrarily close to the nonlinear Dvoretzky phase transition at distortion 2, we thus obtain a much simpler probabilistic proof of the main result of [3], answering a question from [12], and yielding the best known bounds in the nonlinear Dvoretzky theorem. Our method utilizes a sequence of random scales at which a given metric space is fragmented. As in many previous randomized arguments in embedding theory, these scales are chosen irrespective of the geometry of the metric space in question. We show that our bounds are sharp if one utilizes such a "scale-oblivious" fragmentation procedure.

Original languageEnglish (US)
Pages (from-to)489-504
Number of pages16
JournalIsrael Journal of Mathematics
Volume192
Issue number1
DOIs
StatePublished - Dec 2012

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Fragmentation
Metric space
Metric
Theorem
Phase Transition
Target
Subset

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  • Mathematics(all)

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Scale-oblivious metric fragmentation and the nonlinear Dvoretzky theorem. / Naor, Assaf; Tao, Terence.

In: Israel Journal of Mathematics, Vol. 192, No. 1, 12.2012, p. 489-504.

Research output: Contribution to journalArticle

Naor, Assaf ; Tao, Terence. / Scale-oblivious metric fragmentation and the nonlinear Dvoretzky theorem. In: Israel Journal of Mathematics. 2012 ; Vol. 192, No. 1. pp. 489-504.
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