On contact numbers of totally separable unit sphere packings

Contact numbers are natural extensions of kissing numbers. In this paper we give estimates for the number of contacts in a totally separable packing of n unit balls in Euclidean d-space for all n>1 and d>1.Comment: 11 page

On contact graphs of totally separable packings in low dimensions

The contact graph of a packing of translates of a convex body in Euclidean d-space E-d is the simple graph whose vertices are the members of the packing, and whose two vertices are connected by an edge if the two members touch each other. A packing of translates of a convex body is called totally separable, if any two members can be separated by a hyperplane in E-d disjoint from the interior of every packing element. We give upper bounds on the maximum degree (called separable Hadwiger number) and the maximum number of edges (called separable contact number) of the contact graph of a totally separable packing of n translates of an arbitrary smooth convex body in E-d with d = 2, 3, 4. In the proofs, linear algebraic and convexity methods are combined with volumetric and packing density estimates based on the underlying isoperimetric (resp., reverse isoperimetric) inequality. (C) 2018 Elsevier Inc. All rights reserved

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