Contacts in totally separable packings in the plane and in high dimensions

We study the contact structure of totally separable packings of translates of a convex body K in Rd, that is, packings where any two translates of the packing have a separating hyperplane that does not intersect the interior of any translate in the packing. The separable Hadwiger number Hsep(K) of K is defined to be the maximum number of translates touched by a single translate, with the maximum taken over all totally separable packings of translates of K. We show that for each d ≥ 8, there exists a smooth and strictly convex K in Rd with Hsep(K) > 2d, and asymptotically, Hsep(K) = Ω((3/√8)d). We show that Alon’s packing of Euclidean unit balls such that each translate touches at least 2√d others whenever d is a power of 4, can be adapted to give a totally separable packing of translates of the ℓ1-unit ball with the same touching property. We also consider the maximum number of touching pairs in a totally separable packing of n translates of any planar convex body K. We prove that the maximum equals ⌊2n − 2√n⌋ if and only if K is a quasi hexagon, thus completing the determination of this value for all planar convex bodies.