Szlenk indices of convex hulls

We study the general measures of non-compactness defined on subsets of a dual Banach space, their associated derivations and their co-iterates. We introduce the notions of convexifiable and sublinear measure of non-compactness and investigate the properties of its associated fragment and slice derivations. We apply our results to the Kuratowski measure of non compactness and to the study of the Szlenk index of a Banach space. As a consequence, we obtain that the Szlenk index and the convex Szlenk index of a separable Banach space are always equal. We also give, for any countable ordinal a, a characterization of the Banach spaces with Szlenk index bounded by omega(alpha+1) in terms of the existence of an equivalent renorming. This extends a result by Knaust, Odell and Schlumprecht on Banach spaces with Szlenk index equal to omega. (C) 2016 Elsevier Inc. All rights reserved.