Two remarks on the structure of sets of exposed and extreme points (Q2721521)

Let \(K\) be a convex subset of a Banach space \(X\). The element \(x\in K\) is an exposed point of \(K\) if there is a continuous linear form \(f\in X^*\) such that \(\{x\}= \{z\in K:f(z)= \sup_K f\}\). We use \(\exp K\) to denote the set of all exposed points of \(K\). The element \(x\) of \(K\) is an extreme point of \(K\) if \(x\) does not belong to any open segment contained in \(K\). The set of all extreme points of \(K\) is denoted by \(\text{ext }K\) here.NEWLINENEWLINENEWLINEIn the present paper the authors, in section 1, prove the following theorem:NEWLINENEWLINENEWLINETheorem 1. Let \(K\) be a convex subset of a Banach space \(X\) such that \(K\) is compact and separable with respect to the weak topology.NEWLINENEWLINENEWLINE(a) Then \(\text{ext }K\) is a \(K_{\sigma\sigma}\) subset of \(X\) endowed with the weak topology.NEWLINENEWLINENEWLINE(b) Then \(\exp K\) is an \(F_{\sigma\sigma}\) subset of \(X\) endowed with the norm topology. It is also Borel in the weak topology of \(X\). Moreover, it is \(G_{\delta\sigma\delta}\) in the compact metrizable weak topology of \(K\).NEWLINENEWLINENEWLINEThis solves Problem 1.14 from the paper, \textit{G. Choquet}, \textit{H. H. Corson} and \textit{V. Klee} [Pac. J. Math. 17, 33-43 (1966; Zbl 0139.06802)].NEWLINENEWLINENEWLINEIn section 2, the authors show that in many nonseparable Banach spaces there is a bounded closed convex set \(K\) such that set \(\exp K\) and \(\text{ext }K\) are not Borel.