A new metric invariant for Banach spaces

F. Baudier, N. J. Kalton, G. Lancien
2010 Studia Mathematica  
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.
more » ... r uniform homeomorphisms.
doi:10.4064/sm199-1-5 fatcat:bq3ydf6yyfc4jgd3llctqztvbm