### 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 language | English (US) |
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Pages (from-to) | 489-504 |

Number of pages | 16 |

Journal | Israel Journal of Mathematics |

Volume | 192 |

Issue number | 1 |

DOIs | |

State | Published - Dec 2012 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Israel Journal of Mathematics*,

*192*(1), 489-504. https://doi.org/10.1007/s11856-012-0039-7

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

Research output: Contribution to journal › Article

*Israel Journal of Mathematics*, vol. 192, no. 1, pp. 489-504. https://doi.org/10.1007/s11856-012-0039-7

}

TY - JOUR

T1 - Scale-oblivious metric fragmentation and the nonlinear Dvoretzky theorem

AU - Naor, Assaf

AU - Tao, Terence

PY - 2012/12

Y1 - 2012/12

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84871740206&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84871740206&partnerID=8YFLogxK

U2 - 10.1007/s11856-012-0039-7

DO - 10.1007/s11856-012-0039-7

M3 - Article

AN - SCOPUS:84871740206

VL - 192

SP - 489

EP - 504

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

IS - 1

ER -