Denting points of convex sets and weak property \((\pi)\) of cones in locally convex spaces
The authors continue their work on the geometry of sets in a locally convex space \((E, \tau)\). Analogous to the well-studied case of normed linear spaces, the authors first introduce stronger notions of extremality for a convex set \(C\subset E\), like strongly extreme and denting points of a set. It is shown that a denting point is a strongly extreme point. When some of these notions are considered w.r.t. the weak topology \(\sigma(E,E^\ast)\), a strong extreme point \(x \in C\) (assuming that for a neighbourhood \(U\) of \(x\), \(C\cap U\) is a bounded convex set), is then characterized in terms of a family of open slices of \(C \cap U\) containing \(x\), forming a local base for \(x\) in the relative topology on \(C \cap U\). One misses the mention of the work of H. P. Rosenthal in the references.
scientific article; zbMATH DE number 7358925
scientific article