On the closeness of ball packings and the thickness of point sets in the spherical space
type: Item
instance of: scholarly article
title: On the closeness of ball packings and the thickness of point sets in the spherical space
zbMATH Open document ID: 0892.52011
DOI: 10.1023/A:1004228708674
published in: Periodica Mathematica Hungarica
publication date: 1998-03-26T00:00:00Z
review text: The authors study point sets on the unit sphere \(S^d\) (sphere of radius one in \(d+1\)-dimensional space) that admit packings of balls (spherical caps) of angular radius \(r\), equivalently: that have smallest distance \(2r\). Among these they look for packings of minimal closeness (for which the maximal ball that fits in a hole of the packing is as small as possible), and, as a limiting situation, for point sets of maximal thickness (for which the largest empty ball is as small as possible). This is considered as function of the packing radius \(r\), not of the number of points. For big \(r\) (\({\pi\over 2}\geq r\geq{\pi\over 4}\)) the extremal point sets turn out to be vertices of regular simplices in subspaces of increasing dimension, and finally of the crosspolytope. For smaller \(r\) (\(r<{\pi\over 4}\)) bounds are obtained, which are met for the icosahedron and the 600-cell. For the \(S^2\) (in 3-dimensional space) better bounds and additional constructions are given.
Mathematics Subject Classification ID: 52C17
Mathematics Subject Classification ID: 52B11
zbMATH DE Number: 1134652
zbMATH Keywords: close packings of balls
zbMATH Keywords: thick point sets
zbMATH Keywords: spherical space
zbMATH Keywords: spherical caps
zbMATH Keywords: closeness
zbMATH Keywords: thickness
zbMATH Keywords: spherical codes
author: J. Horváth
author: Ágota H. Temesvári
reviewed by: Peter Brass
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1134652