Further investigation of abstract convexity with respect to the class of general min-type functions - Shveidel, A. P. (Optimization, 2007)
In this article we continue the examination of the basic concepts of abstract convexity, taking for the set of elementary functions the totality of the so-called min-type functions which are defined on the n-dimensional Euclidean space R n as the pointwise minimum of finite collections of linear functions. We examine compact -convex sets, -convex hull and -extreme points of a subset of . We show that among global maximizers of a convex-along-rays function over an -convex set there are -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 is -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 .