Some remarks concerning kissing numbers, blocking numbers and covering numbers
type: Item
instance of: scholarly article
title: Some remarks concerning kissing numbers, blocking numbers and covering numbers
zbMATH Open document ID: 0826.52018
DOI: 10.1007/BF01876622
published in: Periodica Mathematica Hungarica
publication date: 1995-11-19T00:00:00Z
review text: The author considers three functionals of convex bodies which are usually investigated in combinatorial geometry. Let \(K\) be a \(d\)-dimensional convex body. The kissing number \(N(K)\) of \(K\) is the largest number of non-overlapping translates of \(K\) which can be brought into contact with \(K\). The blocking number \(Z(K)\) of \(K\) (which was recently introduced by the author) is the smallest number of non-overlapping translates of \(K\) all of which are in contact with \(K\) and which are sufficient to prevent, any other translate of \(K\) from touching \(K\). The covering number \(H(K)\) of \(K\) is the smallest number of translates of \(\text{int} (K)\), the union of which covers the body \(K\). First, the author shows that \(H(K) \leq Z(K)\) for each centrally symmetric convex body \(K\). By the known relation \(N(K) \leq 3^d - 1\) and \(Z(K) \leq N(K)\), this yields \(H(K) \leq 3^d - 1\) if \(K\) has central symmetry, i.e., known upper bounds on \(H(K)\), obtained by Lewin, Petunin, Rogers, and Erdös for centrally symmetric bodies, are improved. Further on, the following phenomenon is proved: when \(d\) is sufficiently large, there are two \(d\)-dimensional convex bodies \(K_1\) and \(K_2\) such that both \(N(K_1) < N(K_2)\) and \(Z(K_1) > Z(K_2)\) hold simultaneously. Finally, the author gives a conjecture regarding the quantity \(Z(K)\), namely \(2d \leq Z(K) \leq 2^d\).
reviewed by: Horst Martini
Mathematics Subject Classification ID: 52C17
Mathematics Subject Classification ID: 05B40
Mathematics Subject Classification ID: 11H31
zbMATH DE Number: 769578
zbMATH Keywords: Hadwiger's covering problem
zbMATH Keywords: illumination problem
zbMATH Keywords: kissing number
zbMATH Keywords: blocking number
author: Chuan Ming Zong
MaRDI profile type: Publication
cites work: Q3138750
cites work: On affine subspaces that illuminate a convex set
cites work: The star number of coverings of space with convex bodies
cites work: Q3751649
description: scientific article; zbMATH DE number 769578