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

Abstract

We study the contact structure of totally separable} packings of translates of a convex body $K$ in $\mathbb{R}^d$, that is, packings where any two touching bodies have a separating hyperplane that does not intersect the interior of any translate in the packing. The separable Hadwiger number $H_{\text{sep}}(K)$ 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 $d\geq 8$, there exists a smooth and strictly convex $K$ in $\mathbb{R}^d$ with $H_{\text{sep}}(K)>2d$, and asymptotically, $H_{\text{sep}}(K)=\Omega\bigl((3/\sqrt{8})^d\bigr)$. We show that Alon's packing of Euclidean unit balls such that each translate touches at least $2^{\sqrt{d}}$ others whenever $d$ is a power of $4$, can be adapted to give a totally separable packing of translates of the $\ell_1$-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 $\lfloor 2n-2\sqrt{n}\rfloor$ if and only if $K$ is a quasi hexagon, thus completing the determination of this value for all planar convex bodies.