Visibility in semi-convex spaces
type: Item
instance of: scholarly article
title: Visibility in semi-convex spaces
zbMATH Open document ID: 0893.52001
DOI: 10.1007/BF01252225
author: Sven Schuierer
author: D. Wood
published in: Journal of Geometry
publication date: 1998-08-24T00:00:00Z
review text: A semi-convex space is a pair \((X,C_X)\) consisting of a set \(X\) and a collection \(C_X\) of subsets of \(X\) such that \(\varnothing\in C_X\) and, for all nonempty \(C \subseteq C_X\), one has \(\cap C\in C_X\) (in contrast with the definition of a convexity space, it is not required that \(X\in C_X)\). The convex hull of \(Y\subseteq X\) is defined as \(C_X- \text{hull} (Y)=\cap \{C\in C_X| Y\subseteq C\}\), with the unusual convention \(\cap \varnothing= \varnothing\). Setting \(C_Y= \{C\in C_X| C\subseteq Y\}\), one says that \(x\) \(C_Y\)-sees \(y\) in \(Y\) if \(C_Y-\text{hull} (\{x,y\}) \neq\varnothing\). One further defines \(C_X\)-kernel\((Y)= \{y\in Y| \text{ for all } y'\in Y,\;y\) \(C_Y\)-sees \(y'\}\). The maximal elements (with respect to inclusion) of \(C_Y\) are called \(C_X\)-skulls of \(Y\). The authors prove several theorems relating skulls and kernels, and study semi-convex spaces in the plane induced by metrics. The theory is illustrated with two examples of semi-convex spaces based on geodesic convexity and staircase convexity.
Mathematics Subject Classification ID: 52A01
Mathematics Subject Classification ID: 52A30
zbMATH DE Number: 1106610
zbMATH Keywords: visibility
zbMATH Keywords: kernel
zbMATH Keywords: skull
zbMATH Keywords: semi-convex space
zbMATH Keywords: convexity space
zbMATH Keywords: geodesic convexity
zbMATH Keywords: staircase convexity
reviewed by: Juan-Enrique Martínez-Legaz
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01252225
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description: scientific article; zbMATH DE number 1106610