The structure of hemispaces in \({\mathbb{R}}^ n\)
type: Item
instance of: scholarly article
title: The structure of hemispaces in \({\mathbb{R}}^ n\)
zbMATH Open document ID: 0656.52004
DOI: 10.1016/0024-3795(83)90135-0
author: Ivan Singer
author: Juan-Enrique Martínez-Legaz
published in: Linear Algebra and its Applications
publication date: 1988-01-01T00:00:00Z
review text: A convex set C of Euclidean n-space \({\mathbb{R}}^ n\) such that \({\mathbb{R}}^ n\setminus C\) is also convex is called a hemispace of \({\mathbb{R}}^ n\). The authors of this extensive paper present several characterizations of hemispaces in terms of linear operators and lexicographical order. They also discuss affine transformations preserving a hemispace. The following separation property is shown: if the intersection of convex sets \(C_ 1\),..., \(C_ p\) is empty, then there exist hemispaces \(H_ 1\supset C_ 1\),..., \(H_ p\supset C_ p\) whose intersection is empty.
Mathematics Subject Classification ID: 52A20
zbMATH DE Number: 4071794
zbMATH Keywords: semispace
zbMATH Keywords: convex half-space
zbMATH Keywords: hemispace
zbMATH Keywords: linear operators
zbMATH Keywords: lexicographical order
reviewed by: Marek Lassak
MaRDI profile type: Publication
full work available at URL: 0024-3795(83)90135-0
OpenAlex ID: W1973873191
cites work: The theory of convex geometries
cites work: Maximal convex sets
cites work: Q5530352
cites work: Q5527606
cites work: Q3940405
cites work: The structure of semispaces
cites work: Q3269413
cites work: Convex half-spaces
cites work: Q3685582
cites work: Exact quasiconvex conjugation
cites work: Lexicographical separation in \({\mathbb{R}}^ n\)
cites work: Lexicographical order, lexicographical index, and linear operators
cites work: Q5539251
cites work: Q3343218
cites work: Q5185609
description: scientific article; zbMATH DE number 4071794