Solution of the problem of combinatorial characterization of the dimension of the kernel of a starshaped set
type: Item
instance of: scholarly article
title: Solution of the problem of combinatorial characterization of the dimension of the kernel of a starshaped set
zbMATH Open document ID: 0831.52003
DOI: 10.1007/BF01224037
author: J. Cel
published in: Journal of Geometry
publication date: 1996-02-20T00:00:00Z
review text: In a real linear space, a nonempty subset \(S\) is starshaped if there exists a point \(x\) in \(S\) such that for any point \(y\) in \(S\), the closed line segment connecting \(x\) and \(y\) lies completely in \(S\). The set of all such points \(x\) is the kernel of \(S\), denoted by \(\ker S\). It is proved that \(\ker S = \cap \{\text{conv} A_z : z \in \text{bdry} S\}\) and if \(S\) is closed and connected, then \(\ker S = \cap \{\text{conv} A_z : z \in \text{slnc} S\}\), where \(\text{bdry} S\) and \(\text{slnc} S\) denote the sets of boundary points and strong local nonconvex points of \(S\), respectively, and \(A_z\) is the set of points \(y\) in \(S\) for which each point in some neighborhood of \(z\) is in the kernel. This yields two Krasnosel'skii-type characterizations for the dimension of \(\ker S\) in \(\mathbb{R}^d\).
Mathematics Subject Classification ID: 52A30
Mathematics Subject Classification ID: 52A37
zbMATH DE Number: 785144
zbMATH Keywords: Krasnosel'skii type theorem
zbMATH Keywords: star shaped set
zbMATH Keywords: visibility
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 785144