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 $\mathbb{H}^n$ ($n=2,3$). We construct in the $2-$ and $3-$dimensional hyperbolic spaces hyp-hor 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 density $\frac{\sqrt{12}}π$ and in $\mathbb{H}^3$ the optimal configuration belongs to the $\{7,3,6\}$ Coxeter tiling with density $\approx 1.27297$ 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 orthosche\-mes $\{p,3,6\}$ $(6< p < 7, ~ p\in \mathbb{R})$ whose density function attains its minimum at parameter $p\approx 6.45962$ with density $\approx 1.26885$. That means that this locally optimal hyp-hor configuration provide smaller covering density than the former determined $\approx 1.27297$ but this hyp-hor packing configuration can not be extended to the entirety of hyperbolic space $\mathbb{H}^3$.