Lattice Coverings of n -Space By Spheres

Source abstract

A lattice ∧ in euclidean n -space, E n is a group of vectors under vector addition generated by n independent vectors, X 1 , X 2 … , X n , 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 (J n r n )/(d(∧)) where J n is the volume of the unit sphere in n -dimensional euclidean space. This density will be denoted by θ n (∧). The density of the most efficient lattice covering of n -space by spheres, θ n is the absolute minimum of θ n (∧) considered as a function from the space of all lattices to the real numbers.