The set of packing and covering densities of convex disks

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 $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$, while the exact shape of $\mathrm{\Omega }$ remains a mystery. Here we describe explicitly a leaf-shaped convex region $\mathrm{\Lambda }$ contained in $\mathrm{\Omega }$ and occupying a good portion of $P$. The sets ${\mathrm{\Omega }}_T$ and ${\mathrm{\Omega }}_L$ 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 ${\mathrm{\Omega }}_T$ and ${\mathrm{\Omega }}_L$ are compact. Furthermore, the sets $\mathrm{\Omega }$, ${\mathrm{\Omega }}_T$ and ${\mathrm{\Omega }}_L$ contain the subsets ${\mathrm{\Omega }}^{\star }$, ${\mathrm{\Omega }}_T^{\star }$ and ${\mathrm{\Omega }}_L^{\star }$ respectively, corresponding to the centrally symmetric convex disks $K$, and our leaf $\mathrm{\Lambda }$ is contained in each of ${\mathrm{\Omega }}^{\star }$, ${\mathrm{\Omega }}_T^{\star }$ and ${\mathrm{\Omega }}_L^{\star }$.