This is an announcement for the paper “A metric interpretation of reflexivity for Banach spaces” by Pavlos Motakis and Thomas Schlumprecht.
Abstract: We define two metrics $d_{1, \alpha}$ and $d_{\infty, \alpha}$ on each Schreier family $S_{\alpha}, \alpha<\omega_1$, 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 $\Phi: S_{\alpha}\rightarrow X$ for which $$cd_{\infty, \alpha}(A, B)\leq |\Phi(A)-\Phi(B)|\leq Cd_{1, \alpha}(A, B)$$ for all $A, B\in S_{\alpha}$. Secondly, we prove for separable and reflexive Banach spaces $X$, and certain countable ordinals $\alpha$ that max$(S_z(X), Sz(X^*))\leq\alpha$ if and only if $S_{\alpha}d_{1,\alpha}$ does not bi-Lipschitzly embed into $X$. Here $S_z(Y)$ denotes the Szlenk index of a Banach space $Y$.
The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1604.07271