A new metric invariant for Banach spaces
Volume 199 / 2010
Studia Mathematica 199 (2010), 73-94
MSC: Primary 46B20; Secondary 46T99.
DOI: 10.4064/sm199-1-5
Abstract
We show that if the Szlenk index of a Banach space $X$ is larger than the first infinite ordinal $\omega $ or if the Szlenk index of its dual is larger than $\omega $, then the tree of all finite sequences of integers equipped with the hyperbolic distance metrically embeds into $X$. We show that the converse is true when $X$ is assumed to be reflexive. As an application, we exhibit new classes of Banach spaces that are stable under coarse-Lipschitz embeddings and therefore under uniform homeomorphisms.
