Sphere Coverage in n Dimensions

This paper presents some theoretical results on the sphere coverage problem in the n-dimensional space. These results refer to the minimal number of spheres, denoted by Nk(a), to cover a cuboid. The first properties outline some theoretical results for the numbers Nk(a), including sub-additivity and monotony on each variable. We use then these results to establish some lower and upper bounds for Nk(a), as well as for the minimal density of spheres to achieve k-coverage. Finally, a computation is proposed to approximate the Nk(a) numbers, and some tables are produced to show them for 2D and 3D cuboids.