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arXiv · 2506.11682

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

Arnasli Yahya·Jenő Szirmai
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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.

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Arnasli Yahya, Jenő Szirmai. 2025-06-13. Decomposition method and upper bound density related to congruent saturated hyperball packings in hyperbolic $n-$space. https://arxiv.org/abs/2506.11682

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