Some remarks on almost locally uniformly rotund points

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 Comment. Math. (Prace Mat.) 44 (2004), pp. 163–186. We also show that such a characterization remains true in reflexive Banach spaces.