Distortion in the finite determination result for embeddings of locally finite metric spaces into Banach spaces

TL;DR

This paper investigates the distortion bounds for embeddings of locally finite metric spaces into Banach spaces, providing new upper bounds and strengthening existing results on the finite determination problem.

Contribution

It establishes improved bounds for the distortion constant D(X) for various classes of Banach spaces, including nested finite-dimensional spaces and spaces without nontrivial cotype.

Findings

D((⊕_{n=1}^∞ X_n)_p) ≤ 1^+ for nested finite-dimensional spaces and all p

D((⊕_{n=1}^∞ ℓ^∞_n)_p) = 1^+ for 1<p<∞

D(X) ≤ 4^+ for Banach spaces with no nontrivial cotype

Abstract

Given a Banach space $X$ and a real number $\alpha\ge 1$, we write: (1) $D(X)\le\alpha$ if, for any locally finite metric space $A$, all finite subsets of which admit bilipschitz embeddings into $X$ with distortions $\le C$, the space $A$ itself admits a bilipschitz embedding into $X$ with distortion $\le \alpha\cdot C$; (2) $D(X)=\alpha^+$ if, for every $\varepsilon>0$, the condition $D(X)\le\alpha+\varepsilon$ holds, while $D(X)\le\alpha$ does not; (3) $D(X)\le \alpha^+$ if $D(X)=\alpha^+$ or $D(X)\le \alpha$. It is known that $D(X)$ is bounded by a universal constant, but the available estimates for this constant are rather large. The following results have been proved in this work: (1) $D((\oplus_{n=1}^\infty X_n)_p)\le 1^+$ for every nested family of finite-dimensional Banach spaces $\{X_n\}_{n=1}^\infty$ and every $1\le p\le \infty$. (2) $D((\oplus_{n=1}^\infty \ell^\infty_n)_p)=1^+$ for $1<p<\infty$. (3) $D(X)\le 4^+$ for every Banach space $X$ with no nontrivial cotype. Statement (3) is a strengthening of the Baudier-Lancien result (2008).