On the 1-density of Unit Ball Covering

Motivated by modern applications like image processing and wireless sensor networks, we consider a variation of the Kepler Conjecture. Given any infinite set of unit balls covering the whole space, we want to know the optimal (lim sup) density of the volume which is covered by exactly one ball (i.e., the maximum such density over all unit ball covers, called the optimal 1-density and denoted as δd, where d is the dimension of the Euclidean space). We prove that in 2D the optimal 1-density δ2 = (3 √ 3− π)/π ≈ 0.6539, which is achieved through a hexagonal covering. In 3D, we show numerically that the dodecahedral covering (dc) achieves a 1-density δ3(dc) ≈ 0.315. We conjecture that this is in fact the optimal 1-density in 3D.