A note on the relationship between the Szlenk and $w^{\ast }$-dentability indices of arbitrary $w^{\ast }$-compact sets

Abstract

We prove the optimal estimate between the Szlenk and $w^{\ast }$-dentability indices of an arbitrary $w^{\ast }$-compact subset of the dual of a Banach space. For a given $w^{\ast }$-compact, convex subset $K$ of the dual of a Banach space, we introduce a two player game the winning strategies of which determine the Szlenk index of $K$. We give applications to the $w^{\ast }$-dentability index of a Banach space and of an operator.