Necessary and sufficient conditions for boundedness of extreme points of unbounded convex sets (Q1100440)

The compactness of the closed convex hull C' of the extreme points of an unbounded line-free closed convex set C in \({\mathbb{R}}^ n\) is investigated. The existence of a terminal extreme point (that is, the end point of an extreme ray) of C is proved to be a necessary condition. If C'\(\neq C\), a sufficient condition, which is also necessary in the case that C' has interior, is the compactness of \(\overline{co}(text C)\), the closed convex hull of the terminal extreme points, together with the inclusions \(C'\subseteq \overline{co}(text C)+L,\) where L ranges over the set of lines generated by the extreme directions of C. Another result states that, if C' is compact and has interior but \(\overline{co}(text C)\) does not have interior, then C' is the intersection of C with one of the closed halfspaces associated with the affine hull of the terminal extreme points of C.