A Krein-Milman type theorem for certain unbounded convex sets

Abstract

Bessaga and Pelczynski [2] demonstrated in 1966 that every closed bounded convex set in a dual separable Banach space is the closed convex hull of its extremal points. Asimow [l] has extended Choquet’s notion of well-capped cones to closed convex sets, and has proved that any well-capped closed convex set is equal to the closed convex hull of its extremal points and extre- ma1 rays. In this investigation we consider the case of a closed (unbounded) convex set in a dual separable Banach space and establish a sufficient condition for such a set to be the closed convex hull of its extremal points and extremal rays. The main result is a more general statement (see Proposition 3.1 and corollaries). The criterion alluded to is the property that the closed convex set be well pseudocapped, a more general property than that of being well capped as in Asimow [I] and Choquet [4]. The concepts of dentable sets and denting points [17] are used to obtain well pseudocapped sets. An example shows that the class of sets for which the main result holds, but for which the theory of Asimow does not, is not vacuous. Another example demonstrates that if the denting condition is omitted, then the result doesn’t hold.