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 . The authors have recently shown  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  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 language||English (US)|
|Number of pages||14|
|Journal||Israel Journal of Mathematics|
|State||Published - 2006|
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