Increasing unions of starlike sets
type: Item
instance of: scholarly article
title: Increasing unions of starlike sets
zbMATH Open document ID: 0614.52007
DOI: 10.1007/BF01221938
published in: Journal of Geometry
publication date: 1987-01-01T00:00:00Z
review text: Let S be a subset of \({\mathbb{R}}^ 2\). For points x in the closure of S (denoted \(cl S)\) and y in S, we say that x sees y via S if and only if the segment (x,y] lies in S. For points x,y in \(cl S,\) y is clearly visible from x via S if and only if there is some neighborhood N of y such that x sees via S each point of \(N\cap S\). The set S is called starlike if and only if there is some point p in cl S such that p sees via S each point of S. The author proves the following theorem: Let \(S\subset {\mathbb{R}}^ 2\). Every three points of \(cl S\) are clearly visible via S from a common point of \(cl S\) if and only if each bounded subset of S may be extended to a starlike set in S. When this occurs, set S is expressible as an increasing union of starlike sets.
Mathematics Subject Classification ID: 52A30
Mathematics Subject Classification ID: 52A35
zbMATH DE Number: 3994425
zbMATH Keywords: increasing union of starlike sets
author: Marilyn Breen
reviewed by: Radu Precup
MaRDI profile type: Publication
cites work: A Characterization Theorem for Bounded Starshaped Sets in the Plane
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description: scientific article; zbMATH DE number 3994425