Ball separation characterization of ball dentability and related properties
Sudeshna Basu, Susmita Seal
TL;DR
The work addresses ball separation in Banach spaces, linking separation by balls to dentability and small-diameter geometry. It develops ball separation characterizations for small-diameter properties via $w^*$-slices and denting/PC/SCS notions, and introduces semi variants to study pointwise dentability-like behavior. A novel $\mathcal{A}$-SCS framework is introduced, showing that density of $\mathcal{A}$-SCS points in $X^*$ yields ball-generated convex sets and BGP-type results, thus unifying and extending existing separation and generation theories. Overall, the paper provides new duality, density, and extension results that connect ball separation, dentability, and ball-generation across a broad class of Banach spaces.
Abstract
In Euclidean spaces, every closed, bounded, convex set can be characterized by two equivalent notions of separation properties. This is not true in general for arbitrary Banach spaces. In this work, we present a ball separation characterization for spaces where the unit ball is dentable. We also explore related properties.