Über scharfe Schattengrenzen und Schnitte konvexer Polytope. (Sharp shadow boundaries and sections of convex polytopes)
type: Item
instance of: scholarly article
title: Über scharfe Schattengrenzen und Schnitte konvexer Polytope. (Sharp shadow boundaries and sections of convex polytopes)
zbMATH Open document ID: 0583.52001
author: Horst Martini
published in: Beiträge zur Algebra und Geometrie
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/138296
review text: In the paper of \textit{P. Kleinschmidt} and \textit{U. Pachner} in Mathematika 27, 58-63 (1980; Zbl 0417.52009) a sharp upper bound for the number of sharp shadow boundaries of convex d-polytopes P under central illumination from q are given; a lower bound is not yet known. (A sharp shadow boundary occurs if and only if the lines of support through q intersect P in exactly one point). Here the author finds sharp upper and lower bounds in the case of parallel illumination. As a tool he uses Grünbaum's projective (d-1)-arrangements. Theorem 1: If \(a_ p(m,d)\) denotes the number of sharp shadow boundaries of a convex polytope with m facet normals under parallel illumination, then \(2^{d-2}(m-d+2) \leq a_ p(m,d) \leq \sum^{d-1}_{i=0}\left( \begin{matrix} m-1\\ i\end{matrix} \right).\) Corollary: A convex polytope is a parallelotope if and only if it has \(2^{d-1}\) sharp shadow boundaries under parallel illumination. Theorem 2: A convex polytope is a simplex if and only if the numbers of sharp shadow boundaries for central and for parallel illumination are equal. Using polarity, the author then formulates the corresponding results about the number of partitions of the vertex set of a convex polytope by hyperplanes. Finally, he points out an interesting connection with the projection body [see \textit{T. Bonnesen} and \textit{W. Fenchel}: ''Theorie der konvexen Körper.'' (1934; Zbl 0008.07708)].
Mathematics Subject Classification ID: 52Bxx
zbMATH DE Number: 3933913
zbMATH Keywords: sections
zbMATH Keywords: sharp shadow boundaries of convex d-polytopes
zbMATH Keywords: central illumination
zbMATH Keywords: arrangements
zbMATH Keywords: parallelotope
zbMATH Keywords: parallel illumination
zbMATH Keywords: simplex
reviewed by: Jonathan Schaer
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3933913