Extreme Points of Certain Convex Subsets of Log-Convex Functions.

Abstract

Abstract : The identification of the extreme points of a convex set is an important problem in mathematics. One reason for identifying the extreme points is that they are, in many instances, the basic building blocks for the convex set. For instance, when the convex set is a compact subset of a locally convex space it is the closed convex hull of its extreme points (Krein-Milman Theorem) and if in addition the set is metrizable, any point in it has an integral representation in terms of the extreme points (Choquet's Theorem). Recently the extreme points for certain types of convex compact sets of probability distributions which occur in reliability theory have been identified. The extreme points for the set of discrete decreasing failure rate distributions are identified without appealing to Choquet's Theorem, give an explicit representation for these distributions in terms of the extreme points. It turns out that techniques employed in this last paper can be used to identify the extreme points for certain convex sets of log-convex functions. The purpose of this paper is to present these results.