  We study the general measures of non-compactness defined on subsets of a dual
Banach space, their associated derivations and their $\omega$-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 $\alpha$, 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 $\omega$.
