Densest packing of equal spheres in hyperbolic space
In hyperbolic space the definition of ``densest packing of equal spheres'' poses problems (not encountered in Euclidean space). The authors put some order into this complex situation: they show that there is a natural meaning for ``optimally dense'' packings in hyperbolic space. Furthermore, they show that only for a special countable set of radii values \(R\) do congruent balls of radius \(R\) have optimally dense packing with high symmetry (again in contrast to Euclidean space). They associate to a packing \(x\) an invariant measure \(\mu_x\) on a space of packings and define an average density \(D(\mu_x)\) to such a measure. This cannot be done for every packing since there are packing examples given by \textit{K. Böröczky} [Mat. Lapok 25(1974), 265-306 (1977; Zbl 0394.52012)] and \textit{G. Fejes Tóth} and \textit{W. Kuperberg} [in Handbook of convex geometry. Vol. B, 799-860 (1993; Zbl 0789.52018)] that have no well-defined density. This paper has a fine bibliography, a survey-type presentation and a (possibly) seminal future.
scientific article; zbMATH DE number 1883353