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

• Published in: Optimization Vol. 56; no. 1-2; pp. 129 - 147
• Main Author: Shveidel, A. P.
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis Group 01.02.2007; Taylor & Francis LLC
• Subjects: Abstract convex set; Bipolar set; Co-radiant set; Convex-along-rays function; Euclidean space; Linear equations; Mathematical functions; Mathematics Subject Classifications 2000: 52A30; Min-type function; Negative polar set; Positive polar set; Radiant set; Studies; Support set
• ISSN: 0233-1934; 1029-4945
• DOI: 10.1080/02331930600816247

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 .