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 Pelczynski, 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 a-measurable for each finite regular Borel measure , 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. Introduction. In 1966 Bessaga and Pelczynski [1] proved that each norm closed and bounded convex subset of a separable dual Banach space is the norm closed convex hull of its extreme points, thus providing a type theorem for a certain collection of noncompact sets. (A more elementary proof of this result has been given in [11].) For compact convex sets (in locally convex spaces) the Krein-Milman theorem can easily be reformulated as an integral representation theorem, using measures which are supported by the closure of the set of extreme points (see for example [13, p. 6]). This formulation has a large number of applications to analysis, probability, etcetera. A much more precise kind of representation theorem is provided by Choquet theory, where the measures (at least in the metrizable case) are supported by the set of extreme points, not just the closure. (See [5] and [2]. For a comprehensive and readable account see [13].) The BessagaPelczynski theorem would probably find greater applicability if it too could be reformulated as an integral representation theorem. As a step towards solving this problem we investigate to what extent the Choquet theory applies to the sets described in the Bessaga-Pelczynski theorem. We take this opportunity to thank Professor R. R. Phelps for directing the author's thesis from which this paper is Received by the editors October 17, 1969. AMS 1970 subject classifications. Primary 46B99; Secondary 46G99, 28A40.