Convex sets, extreme points, and simplexes

Abstract

UDC 513/515+517.948 the central topic of the survey is "Choquet boundary theory," i.e., the sphere of problems pertaining to the representation of infinite-dimensional convex sets in terms of their extreme points.The classical foundation for this topic was set down in the following theorem established in 1940 by M. G. Krein and D. P. Mil'man: A compact convex set in a locally convex Hausdorff space coincides with the closed convex hull of its extreme points, i.e., every point of that set is the center of gravity of a probability measure concentrated on the closure of the extreme points.Interpreted as an integral representation theorem, the Kreirr-Mil'man theorem was later refined by Mil'man [20,21] and Choquet [129][130][131][132]. Choquet further introduced the concept of an infinite-dimensional simplex as a set in which every point has a unique representing "extremal" measure.The Choquet theory has a multiplicity of connections with other branches of mathematics. Within the scope of functional analysis it has bearing on ordered linear spaces, normed algebras (C*-algebras and functional algebras), and operator theory, not to mention ramifications in potential theory, probability theory, ergodic theory, the theory of functions (the problem of moments and completely monotonic functions), extremal problems (best-approximation theory, in particular), and the theory of representations of groups.We include in the present survey papers published in the period from 1967 through 1972. We are concerned exclusively with the "pure" Choquet theory, without delving, as a rule, into its applications to the aforementioned branches of mathematics. Books devoted specifically to Choquet theory have been written by Bauer [74], Phelps [411], Goullet de Rugy [253], Choquet [146], and Alfsen [44]; earlier surveys have been published by Choquet [143], Choquet and Meyer [153], Monna [379], D. A. Edwards [196], and Jacobs [290]. Books by Meyer [361], R. Edwards [29], and Semadeni [465] include sections devoted to the Krein--Mil'man theorem, integral representations, and simplexes. Finally, we mention the surveys of Bauer [76] on integral representations, Alfsen [40] on simplexes, Choquet [47] on conic measures, and D. P. Mil'man [22] on the facial structure of convex sets.We use standard terminology, noting, however, the use of the adjective "semibounded" with reference to a convex set that does not contain a straight line, as well as the interpretation of a "compactum" in the sense of a Hausdorff compact topological space.The following notation is used throughout the article: E is a real locally convex Hausdorff-linear topological space; X is a convex (usually closed) set in E; ~ is an arbitrary compactum; C(~) is the space of real continuous functions on ~; M(~) = C*(~) is the Radon measure space on g; M+(~) is the cone of positive measures; M+(~) is a set of probability measures; in general, L + denotes the positive cone of the linear space L; A(X) and P(X) are the subsets of C(X) formed by, respectively, affine and convex functions; ...

Linear Selections of Superlinear Set-Valued Maps with some Applications to Analysis

“…This easily follows from the fact that the closure of the base of S is not a simplex; see, e. g.,[37] or[39] and also Section 7 below.…”