Decomposition method and upper bound density related to congruent saturated hyperball packings in hyperbolic $n-$space

Abstract

In this paper, we study the problem of hyperball (hypersphere) packings in $n$-dimensional hyperbolic space ($n\ge 4$). We prove that to each $n$-dimensional congruent saturated hyperball packing, there is an algorithm to obtain a decomposition of $n$-dimensional hyperbolic space ${\mathbb{H}}^n$ into truncated simplices. We prove, using the above method and the results of the paper \cite{M94}, that the upper bound of the density for saturated congruent hyperball packings, related to the corresponding truncated tetrahedron cells, is attained in a regular truncated simplex. In 4-dimensional hyperbolic space, we determined this upper bound density to be approximately $0.75864$. Moreover, we deny A.~Przeworski's conjecture \cite{P13} regarding the monotonization of the density function in the $4$-dimensional hyperbolic space.

arXiv admin note: substantial text overlap with arXiv:1812.06785, arXiv:1709.04369, arXiv:1405.0248