Some remarks on almost locally uniformly rotund points

Abstract

We study the relations between different notions of almost locally uniformly rotund points that appear in literature. We show that every non-reflexive Banach space admits an equivalent norm having a point in the corresponding unit sphere which is not almost locally uniformly rotund, and which is strongly exposed by all its supporting functionals. This result is in contrast with a characterization due to P. Bandyopadhyay, D. Huang, and B.-L. Lin from 2004. We also show that such a characterization remains true in reflexive Banach spaces.