On embeddings of locally finite metric spaces into $\ell_p$

TL;DR

This paper investigates the sharpness of bilipschitz embedding results for locally finite metric spaces into , showing that for certain p-values, the known approximation cannot be improved.

Contribution

It proves that for p not equal to 2 or , the existing embedding approximation bounds are optimal and cannot be improved.

Findings

The result is sharp for p , excluding p=2 and p=.

Demonstrates the necessity of the +psilon bound in embeddings.

Clarifies the limitations of embedding locally finite metric spaces into .

Abstract

It is known that if finite subsets of a locally finite metric space $M$ admit $C$-bilipschitz embeddings into $\ell_p$ $(1\le p\le \infty)$, then for every $\epsilon>0$, the space $M$ admits a $(C+\epsilon)$-bilipschitz embedding into $\ell_p$. The goal of this paper is to show that for $p\ne 2,\infty$ this result is sharp in the sense that $\epsilon$ cannot be dropped out of its statement.