Boundaries and generation of convex sets (Q1424107)

From the Krein-Milman theorem follows that every weak-star closed convex set \(K\) in the dual \(X^\ast\) of a Banach space \(X\) is the weak-star closed convex hull of its extreme points \(E\). Following the authors, \(E\) \((W)\)-generates \(K\). As is well-known, for many \(K\) we may generate it from \(E\) by taking the norm-closed convex hull; in this case, \(E\) \((N)\)-generates \(K\). \((W)\) and \((N)\) can of course be defined for any subset \(B\) of \(K\). In this paper, the authors localize a way a subset can generate \(K\) which is properly between \((W)\) and \((N)\): \(B\subset K\) is said to \((I)\)-generate \(K\) if whenever \(B=\bigcup_n C_n\), then \(K=\text{clco}(\bigcup_n w^\ast \text{-clco} \;C_n )\). The fundamental observation is that every boundary \(B\) of \(K\) \((I)\)-generates \(K\). From this, the authors obtain old results of Godefroy and Rodé concerning boundaries. Various results are obtained; we only mention the following theorem: For a separable, non-reflexive Banach space \(X\), the set of bounded linear functionals that do not attain their norm on \(B_X\) is never contained in a proper operator range.