2020/02/23 by Eper, Miklós, Szirmai, Jenő
#52B15 #52C17 #52C22 #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2002.10879
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 ℍn (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√(12)π and in ℍ3 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 orthosche\-mes \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 covering density than the former determined ≈ 1.27297 but this hyp-hor packing configuration can not be extended to the entirety of hyperbolic space ℍ3.