Tight embeddability of proper and stable metric spaces

TL;DR

This paper introduces new embeddability concepts for metric spaces and characterizes when proper subsets of Lp spaces can be embedded into Banach spaces, also improving results on stable metric spaces' embeddings.

Contribution

It defines almost Lipschitz and nearly isometric embeddability, providing characterizations for embeddings of Lp subsets and stable metric spaces into Banach spaces.

Findings

Proper subsets of Lp are almost Lipschitz embeddable iff the target space contains uniformly the ll_p^n's.

Stable metric spaces are nearly isometrically embeddable into reflexive Banach spaces.

The work sharpens a result of N. Kalton on embeddings of stable metric spaces.

Abstract

We introduce the notions of almost Lipschitz embeddability and nearly isometric embeddability. We prove that for $p\in [1,\infty]$, every proper subset of $L_p$ is almost Lipschitzly embeddable into a Banach space $X$ if and only if $X$ contains uniformly the $\ell_p^n$'s. We also sharpen a result of N. Kalton by showing that every stable metric space is nearly isometrically embeddable in the class of reflexive Banach spaces.