Weak$^*$ closures and derived sets for convex sets in dual Banach spaces

Abstract: The paper is devoted to the convex-set counterpart of the theory of weak derived sets initiated by Banach and Mazurkiewicz for subspaces. The main result is the following: For every nonreflexive Banach space X and every countable successor ordinal α, there exists a convex subset A in X such that α is the least ordinal for which the weak derived set of order α coincides with the weak closure of A. This result extends the previously known results on weak derived sets by Ostrovskii (2011) and Silber (2021).