Coverings with horo- and hyperballs generated by simply truncated orthoschemes

Abstract

After having investigated the packings derived by horo-and hyperballs related to simple frustum Coxeter orthoscheme tilings we consider the corresponding covering problems (briefly hyp-hor coverings) in n-dimensional hyperbolic spaces H n (n = 2, 3).We construct in the 2− and 3−dimensional hyperbolic spaces hyphor coverings that are generated by simply truncated Coxeter orthocheme tilings and we determine their thinnest covering configurations and their densities.We prove that in the hyperbolic plane (n = 2) the density of the above thinnest hyp-hor covering arbitrarily approximate the universal lower bound of the hypercycle or horocycle covering densityand in H 3 the optimal configuration belongs to the {7, 3, 6} Coxeter tiling with density ≈ 1.27297 that is less than the previously known famous horosphere covering density 1.280 due to L. Fejes Tóth and K. Böröczky. Moreover, we study the hyp-hor coverings in truncated orthoschemes {p, 3, 6} (6 < p < 7, p ∈ R) whose density function attains its minimum at parameter p ≈ 6.45962 with density ≈ 1.26885. That means that this locally optimal hyp-hor configuration provide smaller