Minkowski circle packings on the sphere
A set of balls is said to be a Minkowski set if none of its elements contains in its interior the centre of another. Replacing in a Minkowski set of balls each ball by a concentric ball of half the radius, one obtains a packing of balls which is called a Minkowski packing. The paper under review concerns Minkowski sets and Minkowski packings of circles (spherical caps) on the sphere. Theorem 1: The density of a Minkowski packing of \(n\geq 3\) circles on the sphere is at most \(n(1-{1\over 2} \sin^{-1} \omega_n)/2\), where \(\omega_n= \pi n/6(n-2)\). Theorem 2: The density of a Minkowski set of \(n\geq 3\) circles on the sphere is at most \(n(1-{1 \over 4}\sin^{-2} \omega_n)\). Both bounds are precise for \(n=3, 4, 6\) and 12, and asymptotically precise for large values of \(n\). The author observes, however, that the problem of the densest Minkowski packing of 5 circles and that of the densest Minkowski arrangement of 5 circles are not equivalent.
scientific article; zbMATH DE number 1334934