Finite sphere packing and sphere covering
type: Item
instance of: scholarly article
title: Finite sphere packing and sphere covering
zbMATH Open document ID: 0658.52007
DOI: 10.1007/BF02187713
author: S. H. Smith
published in: Discrete \& Computational Geometry
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/131062
review text: By a convex body we mean a compact convex subset of Euclidean d-space with non-empty interior. Let k be a given positive integer. A central problem of finite packing and covering is to find (A) the minimum volume of all convex bodies containing a packing of k unit balls, and (B) the maximum volume of all convex bodies that can be covered by k unit balls. L. Fejes Tóth and the third author stated the conjectures that, for \(d\geq 5\), the following two configurations give the solutions to the problems (A) and (B) respectively: The centres of the k balls are equidistantly spaced on a straight line such that (A) the convex hull of the balls has minimum volume or, (B) the balls cover a convex body of maximum volume. In the present paper several partial results are given which support both conjectures. Furthermore, some relations between finite and infinite packings and coverings are considered. For this purpose, the authors introduce a special concept of density suited for packings and coverings with respect to the whole space as well as for finite packings and coverings. Some upper and lower bounds for the volume of a parallel body of a regular simplex are developed which are of independent interest. \{Reviewer's remark: On p. 39 there are three mistakes. G(s) has to be replaced twice by \(V(sQ+B^ d)-V(S_{G(s)}+B^ d)\) (1.7 from below), and \(G(16)=23969\) (1.5 from below).\}
Mathematics Subject Classification ID: 52C17
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 05B40
zbMATH DE Number: 4075951
zbMATH Keywords: packing and covering with spheres
zbMATH Keywords: convex body
zbMATH Keywords: density
reviewed by: August Florian
MaRDI profile type: Publication
cites work: Slices of L. Fejes Tóth's sausage conjecture
cites work: Covering space with equal spheres
cites work: Über dichteste Kreislagerung und dünnste Kreisüberdeckung
cites work: Q5824053
cites work: Q3316129
cites work: On two finite covering problems of Bambah, Rogers, Woods and Zassenhaus
cites work: A combinatorial problem connected with differential equations II
cites work: Über die Einlagerung von Kreisen in einen konvexen Bereich
cites work: Nonlinearity of Davenport-Schinzel sequences and of generalized path compression schemes
cites work: A combinatorial perspective on the non-Radon partitions
cites work: Q4194048
cites work: Some Sphere Packings in Higher Space
cites work: New bounds on the number of unit spheres that can touch a unit sphere in n dimensions
cites work: The Closest Packing of Spherical Caps in n Dimensions
cites work: Q5568974
cites work: On the geometrical moments of skew-regular simplices in hyperspherical space, with some applications in geometry and mathematical statistics
cites work: Q3742257
cites work: Research problem 30
cites work: On the density of finite packings
cites work: Research problems
description: scientific article; zbMATH DE number 4075951