## 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 | American English |
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Pages (from-to) | 111-124 |

Number of pages | 14 |

Journal | Israel Journal of Mathematics |

Volume | 151 |

DOIs | |

State | Published - 2006 |

### Bibliographical note

Funding Information:* Supported in part by a grant from tile Israeli National Science Foundation. ** Supported in part by a grant from the Israeli National Science Foundation. t Supported in part by the Landau Center. Received July 13, 2003