Limitations to Fréchet’s metric embedding method

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, anyn-point metric space contains a subset of size at leastn^1−ε which embeds into ℓ₂ with distortion$O(\frac{{log(2/\varepsilon )}}{\varepsilon })$. 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 largen-point metric spaces, such that the distortion of any Fréchet embedding into ℓ_p on subsets of size at leastn^1/2+ε is Ω((logn)^1/p).