Related papers: Decomposition method and upper bound density relat…
In \cite{Sz17-2} we proved that to each saturated congruent hyperball packing exists a decomposition of $3$-dimensional hyperbolic space $\mathbb{H}^3$ into truncated tetrahedra. Therefore, in order to get a density upper bound for…
In this paper we study the problem of hyperball (hypersphere) packings in $3$-dimensional hyperbolic space. We introduce a new definition of the non-compact saturated ball packings and describe to each saturated hyperball packing, a new…
In earlier works \cite{Sz06-1}, \cite{Sz06-2}, \cite{Sz13-3} and \cite{Sz13-4} we have investigated the densest packings and the least dense coverings by congruent hyperballs (hyperspheres) to the regular prism tilings in $n$-dimensional…
In this paper we study congruent and non-congruent hyperball (hypersphere) packings of the truncated regular tetrahedron tilings. These are derived from the Coxeter simplex tilings $\{p,3,3\}$ $(7\le p \in \mathbb{N})$ and $\{5,3,3,3,3\}$…
In \cite{Sz17-2} we considered hyperball packings in $3$-dimensional hyperbolic space. We developed a decomposition algorithm that for each saturated hyperball packing provides a decomposition of $\HYP$ into truncated tetrahedra. In order…
In this paper we study congruent and non-congruent hyperball (hypersphere) packings of the truncated regular octahedron and cube tilings. These are derived from the Coxeter simplex tilings $\{p,3,4\}$ $(7\le p \in \mathbb{N})$ and…
In this paper we consider ball packings in $4$-dimensional hyperbolic space. We show that it is possible to exceed the conjectured $4$-dimensional realizable packing density upper bound due to L. Fejes T\'oth (Regular Figures, 1964). We…
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…
In $n$-dimensional hyperbolic space $\mathbf{H}^n$ $(n\ge2)$ there are $3$-types of spheres (balls): the sphere, horosphere and hypersphere. If $n=2,3$ we know an universal upper bound of the ball packing densities, where each ball volume…
After having investigated the densest packings by congruent hyperballs to the regular prism tilings in the $n$-dimensional hyperbolic space $\mathbb{H}^n$ ($n \in \mathbb{N}, n \ge 3)$ we consider the dual covering problems and determine…
In this paper we prove a theorem that provides an upper bound for the density of packings of congruent copies of a given convex body in $\mathbb{R}^n$; this theorem is a generalization of the linear programming bound for sphere packings. We…
In hyperbolic space density cannot be defined by a limit as we define it in Euclidean space. We describe the local density bounds for sphere packings and we discuss the different attempts to define optimal arrangements in hyperbolic space.
After the investigation of the congruent and non-congruent hyperball packings related to doubly truncated Coxeter orthoscheme tilings \cite{SzJ1}, we consider the corresponding covering problems. In \cite{MSSz} the authors gave a partial…
The aim of this paper to determine the locally densest horoball packing arrangements and their densities with respect to fully asymptotic tetrahedra with at least one plane of symmetry in hyperbolic 3-space $\bar{\mathbf{H}}^3$ extended…
In this paper we determine new upper bounds for the maximal density of translative packings of superballs in three dimensions (unit balls for the $l^p_3$-norm) and of Platonic and Archimedean solids having tetrahedral symmetry. Thereby, we…
The ball (or sphere) packing problem with equal balls, without any symmetry assumption, in a $3$-dimensional space of constant curvature was settled by B\"or\"oczky and Florian for the hyperbolic space $\HYP$ in \cite{BF64} and by proving…
We prove that the highest density of non-overlapping translates of a given centrally symmetric convex domain relative to its outer parallel domain of given outer radius is attained by a lattice packing in the Euclidean plane. This…
In this paper we study the horoball packings related to the hyperbolic 24 cell in the extended hyperbolic space $\overline{\mathbf{H}}^4$ where we allow {\it horoballs in different types} centered at the various vertices of the 24 cell. We…
We obtain an upper bound to the packing density of regular tetrahedra. The bound is obtained by showing the existence, in any packing of regular tetrahedra, of a set of disjoint spheres centered on tetrahedron edges, so that each sphere is…
After investigating the $3$-dimensional case [35], we continue to address and close the problems of optimal ball and horoball packings in truncated Coxeter orthoschemes with parallel faces that exist in $n$-dimensional hyperbolic space…