Chebyshev sets and ball operators
type: Item
instance of: scholarly article
zbMATH Open document ID: 1303.41014
author: Pedro Martín Jiménez
author: Margarita Spirova
author: Horst Martini
publication date: 2014-10-10T00:00:00Z
full work available at URL: jca21032.htm
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 41A63
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 52A21
zbMATH DE Number: 6353701
zbMATH Keywords: ball hull
zbMATH Keywords: Chebyshev center
zbMATH Keywords: Chebyshev set
zbMATH Keywords: centrable set
MaRDI profile type: Publication
title: Chebyshev sets and ball operators
published in: Journal of Convex Analysis
review text: Let \(K\) be a bounded set in a finite-dimensional real normed linear space. The union of centers of all minimal enclosing balls of \(K\) is called the Chebyshev set of \(K\), and its elements are called Chebyshev centers of \(K\). In this paper, the authors derive some results on Chebyshev sets and Chebyshev centers in real Banach spaces of finite dimension. These results show how Chebyshev sets, ball intersections, ball hulls, and completions of bounded sets are related to each other. It is shown that the Chebyshev set of a bounded set \(K\) always contains the Chebyshev set of some completion of \(K\). For a special class of sets viz. centrable sets, the authors derive a necessary and sufficient condition that the Chebyshev set is a singleton. A complete geometric description of the ball hull of a finite set in the plane is also given, which is useful for algorithmical constructions of the ball hull of such sets.
reviewed by: T. D. Narang
description: scientific article; zbMATH DE number 6353701