Necessary and sufficient conditions for boundedness of extreme points of unbounded convex sets
type: Item
instance of: scholarly article
title: Necessary and sufficient conditions for boundedness of extreme points of unbounded convex sets
zbMATH Open document ID: 0639.52003
DOI: 10.1016/0022-247X(88)90312-5
author: Robert G. Batson
published in: Journal of Mathematical Analysis and Applications
publication date: 1988-01-01T00:00:00Z
review text: The compactness of the closed convex hull C' of the extreme points of an unbounded line-free closed convex set C in \({\mathbb{R}}^ n\) is investigated. The existence of a terminal extreme point (that is, the end point of an extreme ray) of C is proved to be a necessary condition. If C'\(\neq C\), a sufficient condition, which is also necessary in the case that C' has interior, is the compactness of \(\overline{co}(text C)\), the closed convex hull of the terminal extreme points, together with the inclusions \(C'\subseteq \overline{co}(text C)+L,\) where L ranges over the set of lines generated by the extreme directions of C. Another result states that, if C' is compact and has interior but \(\overline{co}(text C)\) does not have interior, then C' is the intersection of C with one of the closed halfspaces associated with the affine hull of the terminal extreme points of C.
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 90C25
zbMATH DE Number: 4042705
zbMATH Keywords: convex sets
zbMATH Keywords: extreme rays
zbMATH Keywords: convex hull
zbMATH Keywords: extreme points
MaRDI profile type: Publication
cites work: Q4104377
cites work: Extensions of Radstrom's lemma with application to stability theory of mathematical programming
cites work: Q5599348
description: scientific article; zbMATH DE number 4042705