A planar 3-convex set is indeed a union of six convex sets
type: Item
instance of: scholarly article
title: A planar 3-convex set is indeed a union of six convex sets
zbMATH Open document ID: 1280.52001
author: Noa Nitzan
author: Micha A. Perles
published in: Discrete \& Computational Geometry
publication date: 2013-06-07T00:00:00Z
review text: For a planar set \(S\), two points \(a,b \in S\) see each other via \(S\) if \([a,b]\) is included in \(S\). A planar set \(S\) is called \(3\)-convex, if for every three points of \(S\), at least two see each other via \(S\). By a result of \textit{F.~A.~Valentine} [Pac. J. Math. 7, 1227--1235 (1957; Zbl 0080.15401)], every closed \(3\)-convex \(S\) is a union of three convex sets. The authors of the paper under review drop the condition that \(S\) is closed and show that \(S\) is a union of (at most) six convex sets (the number six is best possible). It should be noted that the authors complete a result of \textit{M.~Breen} [Pac. J. Math. 53, 43--57 (1974; Zbl 0287.52003)], who claimed this property earlier.
reviewed by: Yurii G. Nikonorov
Mathematics Subject Classification ID: 52A10
zbMATH DE Number: 6171522
zbMATH Keywords: non-convexity
zbMATH Keywords: visually independent
zbMATH Keywords: seeing subset
zbMATH Keywords: invisibility graph
zbMATH Keywords: Valentine's theorem (57')
MaRDI profile type: Publication
full work available at URL: s00454-012-9484-7
OpenAlex ID: W1968179455
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cites work: Q4768277
DOI: 10.1007/S00454-012-9484-7
description: scientific article; zbMATH DE number 6171522