The dimension of the kernel in planar sets which are starshaped via \(\alpha\)-paths
type: Item
instance of: scholarly article
title: The dimension of the kernel in planar sets which are starshaped via \(\alpha\)-paths
zbMATH Open document ID: 0928.52004
author: Marilyn Breen
published in: Archiv der Mathematik
publication date: 1998-01-01T00:00:00Z
review text: Let \(S\) be a plane set and \(\lambda:[p,q]\to S\) a continuous mapping from an interval \([p,q]\subset R\) to \(S:\lambda\) is the path of a curve in \(S\) from \(\lambda(p)\) to \(\lambda(q)\). If \(x,y\in S\), \(x\) sees \(y\) via \(\lambda\) if and only if the curve's range contains \(x\) and \(y\). When this occurs \(\lambda(x,y)\) will denote the subpath of \(\lambda\) from \(x\) to \(y\) and \(\|\lambda(x,y)\|\) will denote its length. A path is said to be an \(\alpha\)-path if, given a positive real number \(\alpha\), \(\|\lambda(x,y)\|\leq\alpha\). A set \(S\) is said to be \(\alpha\)-starshaped if a point \(p\in S\) sees every point \(s\in S\) via \(\alpha\)-paths; then \(\text{Ker}_\alpha S\) will denote the set of all the points of \(S\) for which \(S\) is \(\alpha\)-starshaped. The author proves that if \(S\) is a compact planar set whose boundary is a simple closed locally connected curve and \(\alpha\) is positive real number, the \(\alpha\)-kernel of \(S\) has a non-empty interior if and only if for some positive real number \(\varepsilon\) every three points of \(S\) see via \(\alpha\)-paths in \(S\) some pair of points \(a\) and \(b\) such that \(\text{dist}(a,b)\geq\varepsilon\). Furthermore, an example is exhibited which shows that three is the best possible number. A plane set \(S\) is said to be \(\alpha\)-convex if every two of its points are joined by an \(\alpha\)-path in \(S\). Then if \(S\) is a closed simply connected plane set, \(\text{Ker}_\alpha S\) is the intersection of all the maximal \(\alpha\)-convex subsets of \(S\) and \(\text{Ker}_\alpha S\) is itself \(\alpha\)-convex.
Mathematics Subject Classification ID: 52A30
Mathematics Subject Classification ID: 52A35
zbMATH DE Number: 1214017
zbMATH Keywords: \(\alpha\)-path
zbMATH Keywords: \(\alpha\)-starshaped
zbMATH Keywords: \(\alpha\)-convex
reviewed by: Semin Akdoǧan
MaRDI profile type: Publication
full work available at URL: s000130050244
OpenAlex ID: W2064012059
DOI: 10.1007/S000130050244
description: scientific article; zbMATH DE number 1214017