Related papers: Remarks on soft ball packings in dimensions 2 and …
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…
Dense hard-particle packings are intimately related to the structure of low-temperature phases of matter and are useful models of heterogeneous materials and granular media. Most studies of the densest packings in three dimensions have…
It was conjectured by Ulam that the ball has the lowest optimal packing fraction out of all convex, three-dimensional solids. Here we prove that any origin-symmetric convex solid of sufficiently small asphericity can be packed at a higher…
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…
A packing of translates of a convex domain in the Euclidean plane is said to be totally separable if any two packing elements can be separated by a line disjoint from the interior of every packing element. This notion was introduced by G.…
The aim of this paper is to study lattice-like coverings with congruent translation balls and the packings and coverings with a type of translation cylinders in Sol space related to the fundamental lattices. We introduce the notions of the…
We address the question of which convex shapes, when packed as densely as possible under certain restrictions, fill the least space and leave the most empty space. In each different dimension and under each different set of restrictions,…
We consider the problem of identifying the worst point-symmetric shape for covering n-dimensional Euclidean space with lattice translates. Here we focus on the dimensions where the thinnest lattice covering with balls is known and ask…
Let $L \subset {\Bbb R}^3$ be the union of unit balls, whose centres lie on the $z$-axis, and are equidistant with distance $2d \in [2, 2\sqrt{2}]$. Then a packing of unit balls in ${\Bbb R}^3$ consisting of translates of $L$ has a density…
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…
One of the basic problems in discrete geometry is to determine the most efficient packing of congruent replicas of a given convex set $K$ in the plane or in space. The most commonly used measure of efficiency is density. Several types of…
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 \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…
We give upper bounds for the density of unit ball packings relative to their outer parallel domains and discuss their connection to contact numbers. Also, packings of soft balls are introduced and upper bounds are given for the fraction of…
Determining the minimum density of a covering of $\mathbb{R}^{n}$ by Euclidean unit balls as $n\to\infty$ is a major open problem, with the best known results being the lower bound of $\left(\mathrm{e}^{-3/2}+o(1)\right)n$ by Coxeter, Few…
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 $\{5,3,3,3,3\}$…
The present work surveys problems in $n$-dimensional space with $n$ large. Early development in the study of packing and covering in high dimensions was motivated by the geometry of numbers. Subsequent results, such as the discovery of the…
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…
We formulate the problem of generating dense packings of nonoverlapping, non-tiling polyhedra within an adaptive fundamental cell subject to periodic boundary conditions as an optimization problem, which we call the Adaptive Shrinking Cell…
We consider a smooth Euclidean solid cone endowed with a smooth homogeneous density function used to weight Euclidean volume and hypersurface area. By assuming convexity of the cone and a curvature-dimension condition we prove that the…