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- W3100691144 abstract "Given a Banach space $X$ and a real number $alphage 1$, we write: (1) $D(X)lealpha$ 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 $le alphacdot C$; (2) $D(X)=alpha^+$ if, for every $varepsilon>0$, the condition $D(X)lealpha+varepsilon$ holds, while $D(X)lealpha$ 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 $1le ple infty$. (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)." @default.
- W3100691144 created "2020-11-23" @default.
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- W3100691144 date "2018-02-06" @default.
- W3100691144 modified "2023-10-18" @default.
- W3100691144 title "DISTORTION IN THE FINITE DETERMINATION RESULT FOR EMBEDDINGS OF LOCALLY FINITE METRIC SPACES INTO BANACH SPACES" @default.
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- W3100691144 doi "https://doi.org/10.1017/s0017089518000022" @default.
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