A Helly theorem for intersections of orthogonally starshaped sets
type: Item
instance of: scholarly article
title: A Helly theorem for intersections of orthogonally starshaped sets
zbMATH Open document ID: 1034.52006
author: Marilyn Breen
published in: Archiv der Mathematik
publication date: 2003-11-17T00:00:00Z
review text: A nonempty set in the plane is called an orthogonal polygon if it is a connected union of finitely many rectangles whose edges are parallel to the coordinate axes. By a staircase path we mean a path which is a union of successive ``horizontal'' and ``vertical'' segments. Set \(S\) is called starshaped via staircase paths if there exists a point \(p \in S\) such that for every point \(r \in S\) there is a staircase path in \(S\) which contains \(p\) and \(r\). The main theorem says the following. Let \(\mathcal F\) be a finite family of simply connected orthogonal polygons in the plane. If every three of them have nonempty intersection which is starshaped via staircase paths, then the intersection of all sets from \(\mathcal F\) is a simply connected orthogonal polygon which is starshaped via staircase paths.
Mathematics Subject Classification ID: 52A30
Mathematics Subject Classification ID: 52A35
zbMATH DE Number: 2004966
zbMATH Keywords: starshaped set
zbMATH Keywords: starshaped set via starcase paths
zbMATH Keywords: orthogonal polygon
zbMATH Keywords: Helly's theorem
reviewed by: Marek Lassak
MaRDI profile type: Publication
full work available at URL: s00013-003-4639-7
OpenAlex ID: W2041280154
DOI: 10.1007/S00013-003-4639-7
description: scientific article; zbMATH DE number 2004966