Frechet度量嵌入法的局限性
Limitations to Frechet's Metric Embedding Method
摘要：
Frechet的经典等距嵌入的论点已经发展成为一个主要的工具，在研究度量空间。Frechet嵌入的一个重要例子是Bourgain嵌入。作者最近证明了对于每个e>0，任何n点度量空间都包含一个大小至少为n^（1-e）的子集，它嵌入到l_2中，失真为O（\log（2/e）/e）。我们使用的嵌入是非弗雷歇的，本文的目的是表明这不是巧合。具体地说，对于每个e>0，我们构造任意大的n点度量空间，使得在至少为n^{1/2 + e}的子集上嵌入到l_p中的任何Frechet的失真是Ω（（\log n）^{1/p}）。
英文摘要：
Frechet's classical isometric embedding argument has evolved to become a major tool in the study of metric spaces. An important example of a Frechet embedding is Bourgain's embedding. The authors have recently shown that for every e>0 any n-point metric space contains a subset of size at least n^(1-e) which embeds into l_2 with distortion O(\log(2/e) /e). The embedding we used is non-Frechet, and the purpose of this note is to show that this is not coincidental. Specifically, for every e>0, we construct arbitrarily large n-point metric spaces, such that the distortion of any Frechet embedding into l_p on subsets of size at least n^{1/2 + e} is Ω((\log n)^{1/p}).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
