After the investigation of the congruent and non-congruent hyperball packings related to doubly truncated Coxeter orthoscheme tilings [24], we consider the corresponding covering problems. In [21] the authors gave a partial classification of supergroups of some hyperbolic space groups whose fundamental domains will be integer parts of truncated tetrahedra, and determined the optimal congruent hyperball packing and covering configurations belonging to some of these classes.In this paper we compliment these results with the investigation of the non-congruent covering cases, and the remainig congruent cases. We prove, that between congruent and non-congruent hyperball coverings the thinnest belongs to the {7, 3, 7} Coxeter tiling with density ≈ 1.26829. This covering density is smaller than the conjectured lower bound density of L. Fejes Tóth for coverings with balls and horoballs.We also study the local packing arrangements related to {u, 3, 7} (6 < u < 7, u ∈ R) doubly truncated orthoschemes and the corresponding hyperball coverings. We prove, that these coverings are achieved their minimum density at parameter u ≈ 6.45953 with covering density ≈ 1.26454 which is smaller then the above record-small
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
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