A Krasnosel'skii theorem for orthogonal polygons starshaped via staircase \((n + 1)\)-paths
type: Item
instance of: scholarly article
title: A Krasnosel'skii theorem for orthogonal polygons starshaped via staircase \((n + 1)\)-paths
zbMATH Open document ID: 1298.52008
author: Marilyn Breen
published in: Aequationes Mathematicae
publication date: 2011-02-14T00:00:00Z
review text: Krasnosel'skii showed that a non-empty compact set \(S\) in the plane is starshaped if and only if every three points of \(S\) are visible (along straight line segments in \(S\)) from a common point. The author considers Krasnosel'skii-type results for the case where \(S\) is assumed to be a simply connected orthogonal polygon (orthogonality refers to axis parallel sides), and straight line segments are replaced by staircase paths. In an earlier paper she had shown that a simply connected orthogonal polygon \(S\) is starshaped via staircase paths if and only if every two points of \(S\) are visible via staircase paths in \(S\) from a common point. The present work refines this statement: If \(n\geq1\) is such that every two points of \(S\) are visible via staircase paths of length \(n\) in \(S\) from a common point, then \(S\) is starshaped via staircase paths of length \(n+1\). It is shown that the number two is best possible and that \(n+1\) is best possible for \(n\geq 2\). Moreover, the associated staircase \((n+1)\)-kernel is shown to be staircase path convex with paths of lengths at most \(n+1\).
reviewed by: Markus Kiderlen
Mathematics Subject Classification ID: 52A30
Mathematics Subject Classification ID: 52A35
zbMATH DE Number: 5851435
zbMATH Keywords: orthogonal polygon
zbMATH Keywords: sets starshaped via staircase path
zbMATH Keywords: staircase convex
MaRDI profile type: Publication
full work available at URL: s00010-010-0010-9
OpenAlex ID: W1975292105
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DOI: 10.1007/S00010-010-0010-9
description: scientific article; zbMATH DE number 5851435