Mathematics – Metric Geometry
Scientific paper
2011-12-08
Mathematics
Metric Geometry
14 pages, 3 figures. arXiv admin note: substantial text overlap with arXiv:1105.4315
Scientific paper
In \cite{Sz11} we have generalized the notion of the simplicial density function for horoballs in the extended hyperbolic space $\bar{\mathbf{H}}^n, ~(n \ge 2)$, where we have allowed {\it congruent horoballs in different types} centered at the various vertices of a totally asymptotic tetrahedron. By this new aspect, in this paper we study the locally densest horoball packing arrangements and their densities with respect to totally asymptotic regular tetrahedra in hyperbolic $n$-space $\bar{\mathbf{H}}^n$ extended with its absolute figure, where the ideal centers of horoballs give rise to vertices of a totally asymptotic regular tetrahedron. We will prove that, in this sense, {\it the well known B\"or\"oczky density upper bound for "congruent horoball" packings of $\bar{\mathbf{H}}^n$ does not remain valid for $n\ge4$,} but these locally optimal ball arrangements do not have extensions to the whole $n$-dimensional hyperbolic space. Moreover, we determine an explicit formula for the density of the above locally optimal horoball packings, allowing horoballs in different types.
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