Unions of three starshaped sets in \({\mathbb{R}}^ 2\)
type: Item
instance of: scholarly article
title: Unions of three starshaped sets in \({\mathbb{R}}^ 2\)
zbMATH Open document ID: 0689.52002
DOI: 10.1007/BF01231019
author: Marilyn Breen
published in: Journal of Geometry
publication date: 1989-01-01T00:00:00Z
review text: Two theorems concerning ``visibility'' are proved. Theorem 1: Let S be a compact simply connected set in \({\mathbb{R}}^ 2\). If every boundary point of S is clearly visible via S from one of the three points a, b, c, then S is a union of three starshaped sets whose kernels contain a, b, c, respectively. The result fails when the number three is replaced by four. Theorem 2: Let S be a compact, simply connected set in \({\mathbb{R}}^ 2\), with points a, b, c in S and point x in bdry S. For some spherical neighborhood N of x, assume that every point of \(N\cap S\) sees via S at least one of a, b or c. Then there exists some \(y\in N\cap bdry S\) clearly visible via S from at least one of a, b or c.
Mathematics Subject Classification ID: 52A30
zbMATH DE Number: 4129535
zbMATH Keywords: visibility
zbMATH Keywords: starshaped sets
MaRDI profile type: Publication
cites work: Clear visibility and unions of two starshaped sets
cites work: A Krasnosel'skij-type theorem for unions of two starshaped sets in the plane
cites work: A characterization theorem for certain unions of two starshaped sets in \({\mathbb{R}}^ 2\)
cites work: Q5837707
cites work: Finite Unions of Convex Sets
cites work: Q3954445
cites work: Q5338702
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description: scientific article; zbMATH DE number 4129535