Two Generalizations of Property (II) and their characterization
Sudeshna Basu, Susmita Seal
TL;DR
The paper addresses generalized versions of Property (II) for a compatible collection $\mathcal{A}$ of bounded sets, introducing $\mathcal{A}$-Property (II) and strong $\mathcal{A}$-Property (II). It develops ball-separation characterizations for $\mathcal{A}$-semi PC and strong $\mathcal{A}$-semi PC, and proves that these generalized properties are equivalent to the $\tau_{\mathcal{A}}$-density of the corresponding cones in the dual space $X^*$; specifically, $X$ has $\mathcal{A}$-Property (II) (resp. strong $\mathcal{A}$-Property (II)) if and only if the cone of $\mathcal{A}$-semi PC (resp. strong $\mathcal{A}$-semi PC) of $B_{X^*}$ is dense in $X^*$ under $\tau_{\mathcal{A}}$. The work extends Chen–Lin's $w^*$-PC framework to $\mathcal{A}$-PC variants, providing complete characterizations and linking them to ball-hull representations of convex sets. It includes illustrative examples, such as when $\mathcal{A}$ is all bounded sets or all compact convex sets, connecting to Mazur Intersection Properties and semi $w^*$-PC notions, and clarifies the relations among these generalized PC concepts. This yields a unified approach to generalized MIP-type representations and their dual-density criteria across broad families of bounded sets.
Abstract
The concept of Property (II) was introduced and characterized by Chen and Lin \cite{CL}. In this work, given a compatiable collection of bounded sets $\mathcal{A}$, we provide a complete characterization of two generalizations of Property (II) : (i) every closed convex set $A\in \mathcal{A}$ is an intersection of closed convex hulls of finitely many balls and (ii) for every closed convex set $A\in \mathcal{A}$ and $β\geqslant 0$, $\overline{A+βB_X}$ is an intersection of closed convex hulls of finitely many balls.