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abstract: 'We define two metrics $d_{1,\alpha}$ and $d_{\infty,\alpha}$ on each Schreier family ${\mathcal S}_\alpha$, $\alpha<\omega_1$, with which we prove the following metric characterization of reflexivity of a Banach space $X$: $X$ is reflexive if and only if there is an $\alpha<\omega_1$, so that there is no mapping $\Phi:{\mathcal S}_\alpha\to X$ for which $$cd_{\infty,\alpha}(A,B)\le \|\Phi(A)-\Phi(B)\|\le C  d_{1,\alpha}(A,B) \text{ for all $A,B\in{\mathcal S}_\alpha$.}$$ Secondly, we prove for separable and reflexive Banach spaces $X$, and certain countable ordinals $\alpha$ that $\max({\text{\rm Sz}}(X),{\text{\rm Sz}}(X^*))\le \alpha$ if and only if $({\mathcal S}_\alpha, d_{1,\alpha})$ does not bi-Lipschitzly embed into $X$. Here ${\text{\rm Sz}}(Y)$ denotes the Szlenk index of a Banach space $Y$.'
address:
- 'Department of Mathematics, Texas A&M University, College Station, TX 77843, USA '
- 'Department of Mathematics, Texas A&M University, College Station, TX 77843, USA and Faculty of Electrical Engineering, Czech Technical University in Prague, Zikova 4, 166 27, Prague'
author:
- 'P. Motakis'
- 'Th. Schlumprecht'
title: A metric interpretation of reflexivity for Banach spaces
---

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Introduction and statement of the main results {#S:1}
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In this paper we seek a metric characterization of reflexivity of Banach spaces. By a metric characterization of a property of a Banach space we 