A problem of blocking light rays
type: Item
instance of: scholarly article
title: A problem of blocking light rays
zbMATH Open document ID: 0892.52014
DOI: 10.1023/A:1004974630549
published in: Geometriae Dedicata
publication date: 1998-07-06T00:00:00Z
review text: Let \(K\) be an \(n\)-dimensional convex body with interior \(\text{int} (K)\). Let \(g(K)\) (resp. \(h(K))\) be the smallest number of non-overlapping translates of \(K\) (resp. \(\text{int} (K))\) which block all the light rays starting from \(K\). Denote by \(g^*(K)\) and \(h^*(K)\) similar numbers when the translates are taken from lattice packings of \(K\). The author proves the following results: Theorem 1: For every \(n\)-dimensional convex body \(K\) we have \(g(K)\geq 2n\), where the equality holds if and only if \(K\) is a parallelepiped. The result for \(n=2\) had been obtained by Grünbaum in 1961. Theorem 2: For every \(n\)-dimensional convex body \(K\) we have \(g(K)\leq (8e)^n (n+1)^{n-1} n^{3(n^2-1)/2}\). For every \(n\)-dimensional centrally symmetric convex body \(C\) we have \(g(C)\leq (8e)^n (n+1)^{n-1} n^{n^2-1}\). Theorem 3: When \(n\geq 3\), there is an \(n\)-dimensional convex body \(K\) such that \(g^*(K) <\infty\) but \(h^*(K) =\infty\). Theorem 4: For every two-dimensional convex domain \(D\), \(h^*(D)=12\).
Mathematics Subject Classification ID: 52C17
Mathematics Subject Classification ID: 52B10
zbMATH DE Number: 1089231
zbMATH Keywords: convex body
zbMATH Keywords: affine regular hexagon
zbMATH Keywords: Blaschke's selection theorem
zbMATH Keywords: difference body
zbMATH Keywords: Hausdorff metric
zbMATH Keywords: Mehler's selection theorem
zbMATH Keywords: parallelohedron
zbMATH Keywords: tiling
author: Chuan Ming Zong
reviewed by: Charles Leytem
MaRDI profile type: Publication
full work available at URL: a:1004974630549
OpenAlex ID: W1496556070
description: scientific article; zbMATH DE number 1089231