Related papers: New Horoball Packing Density Lower Bound in Hyperb…
In this paper we study $\times_0$-products of Lann\'er diagrams. We prove that every $\times_0$-product of at least four Lann\'er diagrams with at least one diagram of order $\ge 3$ is superhyperbolic. As a corollary, we obtain that known…
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,…
In \cite{Sz13-1} we defined and described the {\it regular infinite or bounded} $p$-gonal prism tilings in $\SLR$ space. We proved that there exist infinitely many regular infinite $p$-gonal face-to-face prism tilings $\cT^i_p(q)$ and…
Sphere packings in high dimensions interest mathematicians and physicists and have direct applications in communications theory. Remarkably, no one has been able to provide exponential improvement on a 100-year-old lower bound on the…
Let $\mathcal{X}_S$ denote the class of spaces homeomorphic to two closed orientable surfaces of genus greater than one identified to each other along an essential simple closed curve in each surface. Let $\mathcal{C}_S$ denote the set of…
We provide, for any $r\in (0,1)$, lower and upper bounds on the maximal density of a packing in the Euclidean plane of discs of radius $1$ and $r$. The lower bounds are mostly folk, but the upper bounds improve the best previously known…
Packing problems, which ask how to arrange a collection of objects in space to meet certain criteria, are important in a great many physical and biological systems, where geometrical arrangements at small scales control behaviour at larger…
Relatively little is known about the discrete horospheres in hyperbolic groups, even in simple settings. In this paper we work with hyperbolic one-ended right-angled Coxeter groups and describe two graph structures that mimic the intrinsic…
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…
The Hessian of the renormalized volume of geometrically finite hyperbolic $3$-manifolds without rank-$1$ cusps, computed at the hyperbolic metric $g$ with totally geodesic boundary of the convex core, is shown to be a strictly positive…
We analyze Sz\"oll\H{o}si's recent construction of a conjecturally optimal five-dimensional kissing configuration and produce a new such configuration, the fourth to be discovered. We construct five-dimensional sphere packings from these…
This paper provides the currently best known upper bound on the density of a packing in three-dimensional Euclidean space of two types of spheres whose size ratio is the largest one that allows the insertion of a small sphere in each…
For $d=4, 5, 6, 7, 8$, we exhibit examples of $\mathrm{AdS}^{d,1}$ strictly GHC-regular groups which are not quasi-isometric to the hyperbolic space $\mathbb{H}^d$, nor to any symmetric space. This provides a negative answer to Question 5.2…
The packing of hard spheres (HS) of diameter $\sigma$ in a cylinder has been used to model experimental systems, such as fullerenes in nanotubes and colloidal wire assembly. Finding the densest packings of HS under this type of confinement,…
We show that the dynamics of particles in a one-dimensional harmonic trap with hard-core interactions can be solvable for certain arrangements of unequal masses. For any number of particles, there exist two families of unequal mass…
Packing spheres efficiently in large dimension $d$ is a particularly difficult optimization problem. In this paper we add an isotropic interaction potential to the pure hard-core repulsion, and show that one can tune it in order to maximize…
We study the intrinsic shape and alignment of isodensities of galaxy cluster halos extracted from the MultiDark MDR1 cosmological simulation. We find that the simulated halos, are extremely prolate on small scales, and increasingly…
If a hyperbolic 3-manifold admits an exceptional Dehn filling, then the length of the slope of that Dehn filling is known to be at most six. However, the bound of six appears to be sharp only in the toroidal case. In this paper, we…
This review paper is devoted to the problems of sphere packings in 4 dimensions. The main goal is to find reasonable approaches for solutions to problems related to densest sphere packings in 4-dimensional Euclidean space. We consider two…
We investigate the asymptotic best approximation of a smooth, strictly convex body $K$ in $\mathbb{R}^d$ by inscribed polytopes with a restricted number of vertices under the intrinsic volume difference. We prove rigidity phenomena in both…