Limitations to Fréchet’s Metric Embedding Method

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 `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 2 > 0, we construct arbitrarily large n-point metric spaces, such that the distortion of any Fréchet embedding into `p on subsets of size at least n is Ω ( (log n) ) .