On contact graphs of totally separable packings in low dimensions

Contact numbers are natural extensions of kissing numbers. In this paper we give estimates for the number of contacts in a totally separable packing of n unit balls in Euclidean d-space for all n > 1 and d > 1. * Keywords: unit sphere packing, touching pairs, density, (truncated) Voronoi cell, union of balls, isoperimetric inequality, spherical cap packing.

Journal of Geometry, 2013

The contact graph of an arbitrary finite packing of unit balls in Euclidean 3-space is the (simple) graph whose vertices correspond to the packing elements and whose two vertices are connected by an edge if the corresponding two packing elements touch each other. One of the most basic questions on contact graphs is to find the maximum number of edges that a contact graph of a packing of n unit balls can have. In this paper, improving earlier estimates, we prove that the number of touching pairs in an arbitrary packing of n unit balls in E 3 is always less than 6n − 0.926n 2 3 . Moreover, as a natural extension of the above problem, we propose to study the maximum number of touching triplets (resp., quadruples) in an arbitrary packing of n unit balls in Euclidean 3-space. In particular, we prove that the number of touching triplets (resp., quadruples) in an arbitrary packing of n unit balls in E 3 is at most 25 3 n (resp., 11 4 n). *

2003

We show that in d-dimensional Euclidean space the maximum number of non-overlapping translates of a d-dimensional convex body K that can touch K and can lie in a closed supporting half-space of K is always at most 2 • 3 d−1 − 1, with this bound to be reached only if K is an affine image of a d-cube. Such onesided Hadwiger configurations occur at the boundary of finite packings or near the holes in packings of density 0, so this implies that in d-dimensional Euclidean space any k +-neighbour packing by translates of a d-dimensional convex body has positive density for all k ≥ 2•3 d−1 ; and there is a (2•3 d−1 −1) +-neighbour packing by translates of a d-cube that has density 0. (A packing is called a k +-neighbour packing if each packing element has at least k neighbours.) This answers an old question of L. Fejes Tóth (1973).

2018

A packing of translates of a convex body in the d-dimensional Euclidean space E is said to be totally separable if any two packing elements can be separated by a hyperplane of E disjoint from the interior of every packing element. We call the packing P of translates of a centrally symmetric convex body C in E a ρ-separable packing for given ρ ≥ 1 if in every ball concentric to a packing element of P having radius ρ (measured in the norm generated by C) the corresponding sub-packing of P is totally separable. The main result of this paper is the following theorem. Consider the convex hull Q of n non-overlapping translates of an arbitrary centrally symmetric convex body C forming a ρ-separable packing in E with n being sufficiently large for given ρ ≥ 1. If Q has minimal mean i-dimensional projection for given i with 1 ≤ i < d, then Q is approximately a d-dimensional ball. This extends a theorem of K. Böröczky Jr. [Monatsh. Math. 118 (1994), 41–54] from translative packings to ρ-se...

arXiv (Cornell University), 2017

A packing of translates of a convex body in the d-dimensional Euclidean space E d is said to be totally separable if any two packing elements can be separated by a hyperplane of E d disjoint from the interior of every packing element. We call the packing P of translates of a centrally symmetric convex body C in E d a ρ-separable packing for given ρ ≥ 1 if in every ball concentric to a packing element of P having radius ρ (measured in the norm generated by C) the corresponding sub-packing of P is totally separable. The main result of this paper is the following theorem. Consider the convex hull Q of n non-overlapping translates of an arbitrary centrally symmetric convex body C forming a ρ-separable packing in E d with n being sufficiently large for given ρ ≥ 1. If Q has minimal mean i-dimensional projection for given i with 1 ≤ i < d, then Q is approximately a d-dimensional ball. This extends a theorem of K. Böröczky Jr. [Monatsh. Math. 118 (1994), 41-54] from translative packings to ρ-separable translative packings for ρ ≥ 1.

Monatshefte für Mathematik, 2018

A packing of translates of a convex body in the d-dimensional Euclidean space E d is said to be totally separable if any two packing elements can be separated by a hyperplane of E d disjoint from the interior of every packing element. We call the packing P of translates of a centrally symmetric convex body C in E d a ρ-separable packing for given ρ ≥ 1 if in every ball concentric to a packing element of P having radius ρ (measured in the norm generated by C) the corresponding sub-packing of P is totally separable. The main result of this paper is the following theorem. Consider the convex hull Q of n non-overlapping translates of an arbitrary centrally symmetric convex body C forming a ρ-separable packing in E d with n being sufficiently large for given ρ ≥ 1. If Q has minimal mean i-dimensional projection for given i with 1 ≤ i < d, then Q is approximately a d-dimensional ball. This extends a theorem of K. Böröczky Jr. [Monatsh. Math. 118 (1994), 41-54] from translative packings to ρ-separable translative packings for ρ ≥ 1.

Discrete & Computational Geometry, 2016

In [BL] in relation to the unsolved Bang's plank problem (1951) we obtained a lower bound for the sum of relevant measures of cylinders covering a given d-dimensional convex body. In this paper we provide the packing counterpart of these estimates. We also extend bounds to the case of r-fold covering and packing and show a packing analog of Falconer's results ([Fa]).

2010

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 the problem arise depending on the type of isometries allowed for the packing: packing by translates, lattice packing, translates and point reflections, or all isometries. Due to its connections with number theory, crystallography, etc., lattice packing has been studied most extensively. In two dimensions the theory is fairly well developed, and there are several significant results on lattice packing in three dimensions as well. This article surveys the known results, focusing on the most recent progress. Also, many new problems are stated, indicating directions in which future development of the general packing theory in three dimensions seems feasible.

For every fixed graph H, we determine the H-packing number of K n , for all n > n 0 (H). We prove that if h is the number of edges of H, and gcd(H) = d is the greatest common divisor of the degrees of H, then there exists n 0 = n 0 (H), such that for all n > n 0 ,