Staircase kernels for orthogonal \(d\)-polytopes
type: Item
instance of: scholarly article
title: Staircase kernels for orthogonal \(d\)-polytopes
zbMATH Open document ID: 0866.52005
DOI: 10.1007/BF01298451
published in: Monatshefte für Mathematik
publication date: 1997-07-07T00:00:00Z
full work available at URL: https://eudml.org/doc/178733
review text: The author considers special \(d\)-polytopes in connection with a special kind of starshapeness. The result is a generalization of a theorem for 2-polytopes given in a former note by the author [J. Geom. 51, No. 1-2, 22-30 (1994; Zbl 0815.52004)]. \textit{L. Danzer}, \textit{B. Grünbaum} and \textit{V. Klee}, Proc. Symp. Pure Math. 7, 101-180 (1963; Zbl 0132.17401)] had discussed ordinary starshaped sets with the help of visibility via straight line segments. Now, following \textit{R. Motwani}, \textit{A. Raghunathan} and \textit{H. Saran} [J. Comput. Syst. Sci. 40, No. 1, 19-48 (1990; Zbl 0705.68082)], the author uses visibility via staircase paths. An (oriented) path with edges parallel to the coordinate axes will be called a staircase path if no two of its edges have opposite directions. Then for two points \(x\) and \(y\) in a polytope \(S\) one says \(x\) is visible from \(y\) via staircase paths if there is a staircase path in \(S\) which contains both \(x\) and \(y\). Sharper, \(x\) is clearly visible from \(y\) via staircase paths if for some neighborhood \(N\) of \(x\) each point of \(S\) which belongs to \(N\) is visible from \(y\) via staircase paths. The polytope \(S\) in question is an orthogonal \(d\)-polytope, that is a connected union of finitely many convex polytopes, whose edges are parallel to the coordinate axes. Such a polytope is starshaped via staircase paths if for some point \(p\) in \(S\) each point of \(S\) is visible from \(p\) via staircase paths. The set of all such points \(p\) is called the staircase kernel of \(S\). For these staircase kernels of orthogonal \(d\)-polytopes the author gives a simple representation using clearly visibility. Furthermore she shows how to change the result by using only visibility. Namely, for clearly visibility it suffices to consider only those points of \(S\) being points of local nonconvexity, while only visibility needs all the points of the boundary of \(S\).
Mathematics Subject Classification ID: 52A30
Mathematics Subject Classification ID: 52B11
zbMATH DE Number: 920726
zbMATH Keywords: starshaped sets
zbMATH Keywords: staircase kernels
zbMATH Keywords: \(n\)-dimensional polytopes
zbMATH Keywords: visibility
author: Marilyn Breen
reviewed by: Johannes Böhm
MaRDI profile type: Publication
cites work: A Krasnosel'skij theorem for staircase paths in orthogonal polygons
cites work: A Krasnosel'skii-Type Theorem for Points of Local Nonconvexity
cites work: Points of local nonconvexity, clear visibility, and starshaped sets in \(R^ n\)
cites work: Staircase kernels in orthogonal polygons
cites work: Q5556250
cites work: Q3954445
cites work: Covering orthogonal polygons with star polygons: The perfect graph approach
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description: scientific article; zbMATH DE number 920726