### Abstract

Fréchet's classical isometric embedding argument has evolved to become a major tool in the study of metric spaces. An important example of a Fréchet embedding is Bourgain's embedding [4]. The authors have recently shown [2] that for every ε > 0, any n-point metric space contains a subset of size at least n^{1-ε} which embeds into ℓ_{2} with distortion O(log(2/ε)/ε). The embedding used in [2] is non-Fréchet, and the purpose of this note is to show that this is not coincidental. Specifically, for every ε > 0, we construct arbitrarily large n-point metric spaces, such that the distortion of any Fréchet embedding into ℓ_{p} on subsets of size at least n ^{1/2+ε} is Ω((log n)^{1/P}).

Original language | English (US) |
---|---|

Pages (from-to) | 111-124 |

Number of pages | 14 |

Journal | Israel Journal of Mathematics |

Volume | 151 |

DOIs | |

State | Published - 2006 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Israel Journal of Mathematics*,

*151*, 111-124. https://doi.org/10.1007/BF02777357

**Limitations to fréchet's metric embedding method.** / Bartal, Yair; Linial, Nathan; Mendel, Manor; Naor, Assaf.

Research output: Contribution to journal › Article

*Israel Journal of Mathematics*, vol. 151, pp. 111-124. https://doi.org/10.1007/BF02777357

}

TY - JOUR

T1 - Limitations to fréchet's metric embedding method

AU - Bartal, Yair

AU - Linial, Nathan

AU - Mendel, Manor

AU - Naor, Assaf

PY - 2006

Y1 - 2006

N2 - Fréchet's classical isometric embedding argument has evolved to become a major tool in the study of metric spaces. An important example of a Fréchet embedding is Bourgain's embedding [4]. The authors have recently shown [2] that for every ε > 0, any n-point metric space contains a subset of size at least n1-ε which embeds into ℓ2 with distortion O(log(2/ε)/ε). The embedding used in [2] is non-Fréchet, and the purpose of this note is to show that this is not coincidental. Specifically, for every ε > 0, we construct arbitrarily large n-point metric spaces, such that the distortion of any Fréchet embedding into ℓp on subsets of size at least n 1/2+ε is Ω((log n)1/P).

AB - Fréchet's classical isometric embedding argument has evolved to become a major tool in the study of metric spaces. An important example of a Fréchet embedding is Bourgain's embedding [4]. The authors have recently shown [2] that for every ε > 0, any n-point metric space contains a subset of size at least n1-ε which embeds into ℓ2 with distortion O(log(2/ε)/ε). The embedding used in [2] is non-Fréchet, and the purpose of this note is to show that this is not coincidental. Specifically, for every ε > 0, we construct arbitrarily large n-point metric spaces, such that the distortion of any Fréchet embedding into ℓp on subsets of size at least n 1/2+ε is Ω((log n)1/P).

UR - http://www.scopus.com/inward/record.url?scp=33646407302&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33646407302&partnerID=8YFLogxK

U2 - 10.1007/BF02777357

DO - 10.1007/BF02777357

M3 - Article

VL - 151

SP - 111

EP - 124

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

ER -