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 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 $\max(\text{ Sz}(X),\text{ Sz}(X^*))\le \alpha$ if and only if $({\mathcal S}\alpha, d_{1,\alpha})$ does not bi-Lipschitzly embed into $X$. Here $\text{Sz}(Y)$ denotes the Szlenk index of a Banach space $Y$.