This paper presents some theoretical results on the smaller number Nk(a, b) of sensors to achieve k coverage for the rectangular area [0, a] × [0, b]. The first properties show the numbers Nk(a, b) are sub-additive and increasing on each variable. Based on these results, some lower and upper bounds for Nk(a, b) are introduced. The main result of the article proves that the minimal density of sensors to achieve k-coverage is (k) ≤ k/2 improving a previous result of Ammari and Das [2]. Finally, the numbers N1(a, b) are tabled for some small values of a, b.

This paper presents some theoretical results on the smaller number Nk(a, b, c) of sensors to achieve k coverage for the 3D rectangular area [0, a] × [0, b] × [0, c]. The first properties outline some theoretical results for the numbers Nk(a, b, c), including symmetry, subadditivity, and monotony on each variable. We use then these results to establish some lower and upper bounds for Nk(a, b, c). The main contribution proposes a result concerning the minimal density of sensors to achieve k-coverage.

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 $N_k\left(a\right)$, to cover a cuboid. The first properties outline some theoretical results for the numbers $N_k\left(a\right)$, including sub-additivity and monotony on each variable. We use then these results to establish some lower and upper bounds for $N_k\left(a\right)$, as well as for the minimal density of spheres to achieve k-coverage. Finally, a computation is proposed to approximate the $N_k\left(a\right)$ numbers, and some tables are produced to show them for 2D and 3D cuboids.