We define two metrics and on each Schreier family , , with which we prove the following metric characterization of the reflexivity of a Banach space : is reflexive if and only if there is an such that there is no mapping for which Additionally we prove, for separable and reflexive Banach spaces and certain countable ordinals , that if and only if does not bi-Lipschitzly embed into . Here denotes the Szlenk index of a Banach space .
"A metric interpretation of reflexivity for Banach spaces." Duke Math. J. 166 (16) 3001 - 3084, 1 November 2017. https://doi.org/10.1215/00127094-2017-0021