On totally separable packings of equal balls (Q1107831)

In Euclidean n-space a set of bodies \(\{b_ i\}\) is said to be totally separable if to any pair of bodies \(b_ i\) and \(b_ j\) there is an (n- 1)-dimensional plane p such that \(b_ i\) and \(b_ j\) lie on different sides of p, and p does not intersect the interior of any body \(b_ k\). Theorem: If a 3-dimensional cube of volume V contains a totally separable set of N balls of radius r then \(V\geq 8Nr\) 3. This implies that in 3- space the density of a densest totally separable packing of equal balls is \(\pi\) /6.