Decomposition method and upper bound density related to congruent saturated hyperball packings in hyperbolic $n-$space
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arXiv
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| Format: | Preprint |
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2025
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| author | Yahya, Arnasli Szirmai, Jenő |
| author_facet | Yahya, Arnasli Szirmai, Jenő |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_11682 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Decomposition method and upper bound density related to congruent saturated hyperball packings in hyperbolic $n-$space Yahya, Arnasli Szirmai, Jenő Metric Geometry 52C17, 52C22, 52B15 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. |
| title | Decomposition method and upper bound density related to congruent saturated hyperball packings in hyperbolic $n-$space |
| topic | Metric Geometry 52C17, 52C22, 52B15 |
| url | https://arxiv.org/abs/2506.11682 |