The Packing of Spheres in the Space lp

Abstract

A point x in the real or complex space lpis an infinite sequence,(x1, x2, x3,…) of real or complex numbers such that is convergent. Here p ≥ 1 and we writeThe unit sphere S consists of all points x ε lp for which ¶ x ¶ ≤ 1. The sphere of radius a≥ ≤ 0 and centre y is denoted by Sa(y) and consists of all points x ε lp such that ¶ x - y ¶ ≤ a. The sphere Sa(y) is contained in S if and only if ¶ y ¶≤1 - a, and the two spheres Sa(y) and Sa(z) do not overlap if and only if¶ y- z ¶≥ 2aThe statement that a finite or infinite number of spheres Sa (y) of fixed radius a can be packed in S means that each sphere Sa (y) is contained in S and that no two such spheres overlap.