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

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).