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We generalize the classical notion of packing a set by balls with identical radii to the case where the radii may be different. The largest number of such balls that fit inside the set without overlapping is called its {em non-uniform packing number}. We show that the non-uniform packing number can be upper-bounded in terms of the {em average} radius of the balls, resulting in bounds of the familiar classical form.
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A contact graph of a packing of closed balls is a graph with balls as vertices and pairs of tangent balls as edges. We prove that the average degree of the contact graph of a packing of balls (with possibly different radii) in $mathbb{R}^3$ is not greater than $13.92$. We also find new upper bounds for the average degree of contact graphs in $mathbb{R}^4$ and $mathbb{R}^5$.
Suppose one has a collection of disks of various sizes with disjoint interiors, a packing, in the plane, and suppose the ratio of the smallest radius divided by the largest radius lies between $1$ and $q$. In his 1964 book textit{Regular Figures} (MR0165423), Laszlo Fejes Toth found a series of packings that were his best guess for the maximum density for any $1> q > 0.2$. Meanwhile Gerd Blind in (MR0275291,MR0377702) proved that for $1ge q > 0.72$, the most dense packing possible is $pi/sqrt{12}$, which is when all the disks are the same size. In (MR0165423), the upper bound of the ratio $q$ such that the density of his packings greater than $pi/sqrt{12}$ that Fejes Toth found was $0.6457072159..$. Here we improve that upper bound to $0.6585340820..$. Our new packings are based on a perturbation of a triangulated packing that have three distinct sizes of disks, found by Fernique, Hashemi, and Sizova, (MR4292755), which is something of a surprise.
In this paper we construct a new family of lattice packings for superballs in three dimensions (unit balls for the $l^p_3$ norm) with $p in (1, 1.58]$. We conjecture that the family also exists for $p in (1.58, log_2 3 = 1.5849625ldots]$. Like in the densest lattice packing of regular octahedra, each superball in our family of lattice packings has $14$ neighbors.
We consider packings of congruent circles on a square flat torus, i.e., periodic (w.r.t. a square lattice) planar circle packings, with the maximal circle radius. This problem is interesting due to a practical reason - the problem of super resolution of images. We have found optimal arrangements for N=6, 7 and 8 circles. Surprisingly, for the case N=7 there are three different optimal arrangements. Our proof is based on a computer enumeration of toroidal irreducible contact graphs.
In 1969, Fejes Toth conjectured that in Euclidean 3-space any packing of equal balls such that each ball is touched by twelve others consists of hexagonal layers. This article verifies this conjecture.
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