Further investigation of abstract convexity with respect to the class of general min-type functions

摘要

In this article we continue the examination of the basic concepts of abstract convexity, taking for the set of elementary functions the totality F of the so-called min-type functions which are defined on the n-dimensional Euclidean space R{sup}n as the pointwise minimum of finite collections of linear functions. We examine compact F-convex sets, F-convex hull and F-extreme points of a subset of R{sub}n. We show that among global maximizers of a convex-along-rays function over an F-convex set there are F-extreme points of the set. We introduce the positive and the negative polar sets and examine their properties. We present the description of the bipolar sets and show that a subset of R{sub}n is F-convex if it is the intersection of its positive and negative bipolar sets. We describe the support set of the upper envelope of a subset of F.