Some remarks on almost locally uniformly rotund points
Carlo Alberto De Bernardi, Jacopo Somaglia
TL;DR
The paper investigates the relationship between almost locally uniformly rotund points ($aLUR$) and nicely strongly exposed points ($NSE$) in Banach spaces under renormings. It establishes that non-reflexive spaces admit renormings with an $NSE$ point that fails to be $aLUR$, while in reflexive spaces $NSE$ and $aLUR$ coincide for all renormings. A key consequence is a reflexivity characterization: a Banach space is reflexive iff, for every renorming, the $aLUR$-points of the unit sphere coincide with the $NSE$-points. The authors also provide a double-limit (Pringsheim-sense) criterion for $NSE$, connecting these geometric properties to limit behavior. Overall, the work clarifies the interplay of rotundity notions in renorming theory and corrects previous gaps in the literature regarding the NSE–$aLUR$ equivalence.
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.