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

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,forevery$\varepsilon>0$, the condition $D(X)\le \alpha +\epsilon$ 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) $D(({\oplus }_{n=1}^{\mathrm{\infty }}{X}_n{)}_p)\le 1^{+}$ for every nested family of finite-dimensional Banach spaces $\{X_n{\}}_{n=1}^{\mathrm{\infty }}$ and every $1\le p\le \infty$\mathrm{.}(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).