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Lattice Coverings of n-Space By Spheres

Published online by Cambridge University Press:  20 November 2018

M. N. Bleicher
M. N. Bleicher*
Affiliation:
University of California
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A lattice ∧ in euclidean n-space, En is a group of vectors under vector addition generated by n independent vectors, X1, X2 … , Xn, called a basis for the lattice. The absolute value of the n × n determinant the rows of which are the co-ordinates of a basis is called the determinant of the lattice and is denoted by d(∧). For any lattice ∧ there is a unique minimal positive number r such that, if spheres of radius r are placed with centres at all points of ∧, the entire space is covered. The density of this covering may be defined as (Jnrn)/(d(∧)) where Jn is the volume of the unit sphere in n-dimensional euclidean space. This density will be denoted by θn(∧). The density of the most efficient lattice covering of n-space by spheres, θn is the absolute minimum of θn(∧) considered as a function from the space of all lattices to the real numbers.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 14 , 1962 , pp. 632 - 650
Copyright
Copyright © Canadian Mathematical Society 1962

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