Coverings with horo- and hyperballs generated by simply truncated orthoschmes

Miklós Eper ; Jenő Szirmai 

Novi Sad Journal of Mathematics, Tome 53 (2023) no. 1.

Résumé

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 $\HYN$ ($n=2,3$). In the $2-$ and $3-$dimensional hyperbolic spaces we construct hyp-hor coverings generated by simply truncated Coxeter orthochemes, 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 approximates the universal lower bound of the hypercycle or horocycle covering density $\frac{\sqrt{12}}{\pi}$, and in $\HYP$ 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 T\'oth and K.~B\"or\"oczky. 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 entire hyperbolic space $\m1athbb{H}^3$.

Détail

Publié le : 2021-08-23

DOI : 10.30755/NSJOM.12680

Classification : 53A35, 52C17
Keywords: hyperbolic geometry, horoball and hyperball coverings, covering density, Coxeter tilings