The set of packing and covering densities of convex disks

TL;DR

This paper investigates the set of all possible pairs of packing and covering densities for convex disks in the plane, describing a specific leaf-shaped region within the known bounds and exploring related subsets for symmetric cases.

Contribution

It explicitly characterizes a leaf-shaped convex region within the set of all packing and covering density pairs for convex disks, advancing understanding of their geometric relationships.

Findings

Identifies a leaf-shaped convex region within the density set

Shows the sets are contained within a convex polygon P

Establishes the affine invariance and compactness of related subsets

Abstract

For every convex disk $K$ (a convex compact subset of the plane, with non-void interior), the packing density $\delta(K)$ and covering density $\vartheta(K)$ form an ordered pair of real numbers, {\em i.e.}, a point in ${\mathbb R}^2$. The set $\Omega$ consisting of points assigned this way to all convex disks is the subject of this article. A few known inequalities on $\delta(K)$ and $\vartheta(K)$ jointly outline a relatively small convex polygon $P$ that contains $\Omega$, while the exact shape of $\Omega$ remains a mystery. Here we describe explicitly a leaf-shaped convex region $\Lambda$ contained in $\Omega$ and occupying a good portion of $P$. The sets $\Omega_T$ and $\Omega_L$ of translational packing and covering densities and lattice packing and covering densities are defined similarly, restricting the allowed arrangements of $K$ to translated copies or lattice arrangements, respectively. Due to affine invariance of the translative and lattice density functions, the sets $\Omega_T$ and $\Omega_L$ are compact. Furthermore, the sets $\Omega$, $\Omega_T$ and $\Omega_L$ contain the subsets $\Omega^\star$, $\Omega_T^\star$ and $\Omega_L^\star$ respectively, corresponding to the centrally symmetric convex disks $K$, and our leaf $\Lambda$ is contained in each of $\Omega^\star$, $\Omega_T^\star$ and $\Omega_L^\star$.