A new metric invariant for Banach spaces

Florent Baudier (1) , Nigel Kalton (2) , Gilles Lancien (1)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
2 Department of Mathematics, University of Missouri

We show that if the Szlenk index of a Banach space X or of its dual is larger than the first infinite ordinal, 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.

Domaines: Analyse fonctionnelle [math.FA]

Florent Baudier, Nigel Kalton, Gilles Lancien. A new metric invariant for Banach spaces. Studia Mathematica, 2010, 199, pp.73-94. ⟨hal-00471226⟩