Related papers: New Lower Bound for the Optimal Ball Packing Densi…

We show that in any $d$-dimensional real normed space, unit balls can be packed with density at least \[\frac{(1-o(1))d\log d}{2^{d+1}},\] improving a result of Schmidt from 1958 by a logarithmic factor and generalizing the recent result of…

In this paper we prove an asymptotic lower bound for the sphere packing density in dimensions divisible by four. This asymptotic lower bound improves on previous asymptotic bounds by a constant factor and improves not just lower bounds for…

We investigate lower bounds for the number of ideal and finite vertices of right-angled hyperbolic polyhedra of finite volume. We use a geometric method of orthogonal gluings to establish new bounds in low dimensions, specifically…

After having investigated the regular prisms and prism tilings in the $\SLR$ space in the previous work \cite{Sz13-1} of the second author, we consider the problem of geodesic ball packings related to those tilings and their symmetry groups…

After having investigated several types of geodesic ball packings in $\mathbf{S}^2 \times \mathbf{R}$ space, in this paper we study the locally optimal geodesic of simply and multiply transitive ball packings with equal balls to the space…

We prove sphere packing density bounds in hyperbolic space (and more generally irreducible symmetric spaces of noncompact type), which were conjectured by Cohn and Zhao and generalize Euclidean bounds by Cohn and Elkies. We work within the…

The study of rod complements is motivated by rod packing structures in crystallography. We view them as complements of links comprised of Euclidean geodesics in the 3-torus. Recent work of the second author classifies when such rod…

In 1900, as a part of his 18th problem, Hilbert proposed the question to determine the densest congruent (or translative) packings of a given solid, such as the unit ball or the regular tetrahedron of unit edges. Up to now, our knowledge…

In this paper, we obtain a complete classification of 331 finite-volume hyperbolic Coxeter 4-dimensional polytopes with 7 facets.

We prove that for all fixed $p > 2$, the translative packing density of unit $\ell_p$-balls in $\mathbb{R}^n$ is at most $2^{(\gamma_p + o(1))n}$ with $\gamma_p < - 1/p$. This is the first exponential improvement in high dimensions since…

We study translative arrangements of centrally symmetric convex domains in the plane (resp., of congruent balls in the Euclidean $3$-space) that neither pack nor cover. We define their soft density depending on a soft parameter and prove…

Since there is no hyperbolic Dehn filling theorem for higher dimensions, it is challenging to construct explicit hyperbolic manifolds of small volume in dimension at least four. Here, we build up closed hyperbolic 4-manifolds of volume…

For any given natural number $k$, this paper gives upper bounds on the radius of a packing of a complete hyperbolic surface of finite area by $k$ equal-radius disks in terms of the surface's topology. We show that the bounds given here are…

We study the problem of high-dimensional multiple packing in Euclidean space. Multiple packing is a natural generalization of sphere packing and is defined as follows. Let $ N>0 $ and $ L\in\mathbb{Z}_{\ge2} $. A multiple packing is a set…

Let $X_1,\ldots,X_n$ be independent random points in the closed unit ball of $\mathbb{R}^d$. Assume that each $X_i$ has a beta distribution with parameter $\beta_i \ge -1$: if $\beta_i>-1$, then $X_i$ has Lebesgue density proportional to…

In this paper we get an explicit lower bound for the radius of a Bergman ball contained in the Dirichlet fundamental polyhedron of a torsion-free discrete group $G\subset PU(n,1)$ acting on complex hyperbolic space. Consequently the volume…

We obtain strong upper bounds for the Betti numbers of compact complex-hyperbolic manifolds. We use the unitary holonomy to improve the results given by the most direct application of the techniques of [DS17]. We also provide effective…

We prove a lower bound on the entropy of sphere packings of $\mathbb R^d$ of density $\Theta(d \cdot 2^{-d})$. The entropy measures how plentiful such packings are, and our result is significantly stronger than the trivial lower bound that…

The average distance of the equal hard spheres is introduced to evaluate the density of a given arrangement. The absolute smallest value is two radii because the spheres can not be closer to each other than their diameter. The absolute…

The note shows an easy way to improve E.H. Smith's packing density bound in $\mathbb{R}^3$ from $0.53835...$ to $0.54755...$ .