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Packings with large minimum kissing numbers For each proper power of 4, n, we describe a simple explicit construction of a finite collection
 

Summary: Packings with large minimum kissing numbers
Noga Alon
Abstract
For each proper power of 4, n, we describe a simple explicit construction of a finite collection
of pairwise disjoint open unit balls in Rn
in which each ball touches more than 2

n
others.
A packing of balls in the Euclidean space is a finite or infinite collection of pairwise disjoint open
unit balls in Rn. It is called a lattice packing if the centers of the balls form a lattice in Rn. The
minimum kissing number of a packing is the minimum number of balls touching a given one. Note
that for a lattice packing this is simply the number of balls touching any given one, since every ball
touches the same number of others. The problem of existence and construction of lattice packings
with high kissing numbers received a considerable amount of attention, and there are several known
constructions that show that the kissing number of a lattice packing of balls in Rn may be at least
n(log n) = 2(log2
n). See [3], [4], [2], [6], and [5]. The problem of constructing finite packings with
a large minimum kissing number received much less attention. In this short note we consider this
problem and construct finite packings in Rn with much higher kissing numbers than those of the

  

Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University

 

Collections: Mathematics