On the closeness of ball packings and the thickness of point sets in the spherical space
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.
scientific article; zbMATH DE number 1134652