Low distortion embeddings into Asplund Banach spaces

Abstract

We give a simple example of a countable metric space $M$ that does not embed bi-Lipschitz with distortion strictly less than 2 into any Asplund space. Actually, if $M$ embeds with distortion strictly less than 2 to a Banach space $X$, then $X$ contains an isomorphic copy of ${\ell }_1$. We also show that the space $M$ does not embed with distortion strictly less than $2$ into ${\ell }_1$ itself but it does embed isometrically into a space that is isomorphic to ${\ell }_1$.

arXiv comment5 pages

References in corpus (1)

• A topological obstruction for small-distortion embeddability into spaces of continuous functions on countable compact metric spaces

Cited by in corpus (3)

• Linear properties of Banach spaces and low distortion embeddings of metric graphs
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