A metric interpretation of reflexivity for Banach spaces

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 the reflexivity of a Banach space $X$: $X$ is reflexive if and only if there is an $\alpha <{\omega }_1$ such that there is no mapping $\Phi :{\mathcal{S}}_{\alpha }\to X$ for which $$c{d}_{\infty ,\alpha }(A,B)\le \Vert \Phi \left(A\right)-\Phi \left(B\right)\Vert \le C{d}_{1,\alpha }(A,B)\text{for all}A,B\in {\mathcal{S}}_{\alpha }.$$ Additionally we prove, for separable and reflexive Banach spaces $X$ and certain countable ordinals $\alpha$, that $max\hspace{0.17em}(Sz(X),Sz(X^{\ast }\left)\right)\le \alpha$ if and only if $({\mathcal{S}}_{\alpha },d_{1,\alpha })$ does not bi-Lipschitzly embed into $X$. Here $Sz\left(Y\right)$ denotes the Szlenk index of a Banach space $Y$.