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

TL;DR

This paper investigates hyperball packings in hyperbolic spaces of dimension four and higher, providing an algorithm for space decomposition, establishing an upper density bound, and refuting a previous conjecture about density monotonicity.

Contribution

It introduces an algorithm for decomposing hyperbolic space into truncated simplices and determines the maximal packing density in 4D hyperbolic space, challenging existing conjectures.

Findings

Upper bound density in 4D hyperbolic space is approximately 0.75864.

Decomposition into truncated simplices is possible for all congruent saturated hyperball packings.

The conjecture on density monotonicity in 4D hyperbolic space is disproved.

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.