Barycenters of Measures on Certain Noncompact Convex Sets

Abstract

Each norm closed and bounded convex subset K of a separable dual Banach space is, according to a theorem of Bessaga and Pełczynski, the norm closed convex hull of its extreme points. It is natural to expect that this theorem may be reformulated as an integral representation theorem, and in this connection we have examined the extent to which the Choquet theory applies to such sets. Two integral representation theorems are proved and an example is included which shows that a sharp result obtains for certain noncompact sets. In addition, the set of extreme points of K is shown to be $\mu$-measurable for each finite regular Borel measure $\mu$, hence eliminating certain possible measure-theoretic difficulties in proving a general integral representation theorem. The last section is devoted to the study of a class of extreme points (called pinnacle points) which share important geometric properties with extreme points of compact convex sets in locally convex spaces. A uniqueness result is included for certain simplexes all of whose extreme points are pinnacle points.