Asymptotically convex Banach spaces and The Index of Rotundity Problem

the index of rotundity problem asks whether a banach space which admits equivalent renormings with index of rotundity as small as desired also admits an equivalent rotund renorming. in this paper we continue the ongoing search for a negative answer to this question by making use of a new concept: asymptotically convex banach spaces. some applications to the approximation hyperplane series property are given.