A new metric invariant for Banach spaces

TL;DR

This paper introduces a new metric invariant for Banach spaces based on the Szlenk index, linking it to the embeddability of certain trees and revealing stability properties under coarse-Lipschitz embeddings.

Contribution

It establishes a novel metric invariant related to the Szlenk index and characterizes embeddability of hyperbolic trees, with implications for stability under coarse-Lipschitz maps.

Findings

Szlenk index > ω implies hyperbolic tree embeds into X

Converse holds for reflexive spaces

Identifies new classes of Banach spaces stable under coarse-Lipschitz embeddings

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.