Limitations to fréchet's metric embedding method

Yair Bartal, Nathan Linial, Manor Mendel, Assaf Naor

Research output: Contribution to journalArticle

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 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).

Original languageEnglish (US)
Pages (from-to)111-124
Number of pages14
JournalIsrael Journal of Mathematics
Volume151
DOIs
StatePublished - 2006

Fingerprint

Metric Embeddings
Metric space
Isometric Embedding
Subset

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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

In: Israel Journal of Mathematics, Vol. 151, 2006, p. 111-124.

Research output: Contribution to journalArticle

Bartal, Y, Linial, N, Mendel, M & Naor, A 2006, 'Limitations to fréchet's metric embedding method', Israel Journal of Mathematics, vol. 151, pp. 111-124. https://doi.org/10.1007/BF02777357
Bartal, Yair ; Linial, Nathan ; Mendel, Manor ; Naor, Assaf. / Limitations to fréchet's metric embedding method. In: Israel Journal of Mathematics. 2006 ; Vol. 151. pp. 111-124.
@article{99aee3ff702342999fd554e64ef674b6,
title = "Limitations to fr{\'e}chet's metric embedding method",
abstract = "Fr{\'e}chet's classical isometric embedding argument has evolved to become a major tool in the study of metric spaces. An important example of a Fr{\'e}chet 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-Fr{\'e}chet, 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 Fr{\'e}chet embedding into ℓp on subsets of size at least n 1/2+ε is Ω((log n)1/P).",
author = "Yair Bartal and Nathan Linial and Manor Mendel and Assaf Naor",
year = "2006",
doi = "10.1007/BF02777357",
language = "English (US)",
volume = "151",
pages = "111--124",
journal = "Israel Journal of Mathematics",
issn = "0021-2172",
publisher = "Springer New York",

}

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 -