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

TL;DR

This paper investigates the maximum contact numbers in totally separable packings of convex bodies in high dimensions and the plane, revealing new bounds and characterizations for specific convex shapes.

Contribution

It establishes the existence of convex bodies with exponentially large separable Hadwiger numbers in high dimensions and characterizes maximum contact pairs in planar convex packings.

Findings

Existence of smooth, strictly convex bodies with $H_{sep}(K) > 2d$ for $d \\geq 8$

Asymptotic growth of $H_{sep}(K)$ as $\\Omega((3/\\sqrt{8})^d)$

Maximum contact pairs in planar convex bodies is $\\lfloor 2n-2\\sqrt{n}\\rfloor$ iff $K$ is a quasi hexagon

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.