The Set of Packing and Covering Densities of Convex Disks

For every convex disk $$K$$K (a convex compact subset of the plane, with non-void interior), the packing density $$\delta (K)$$ź(K) and covering density $${\vartheta (K)}$$ź(K) form an ordered pair of real numbers, i.e., a point in $$\mathbb{R }^2$$R2. The set $$\varOmega $$Ω consisting of points assigned this way to all convex disks is the subject of this article. A few known inequalities on $$\delta (K)$$ź(K) and $${\vartheta (K)}$$ź(K) jointly outline a relatively small convex polygon $$P$$P that contains $$\varOmega $$Ω, while the exact shape of $$\varOmega $$Ω remains a mystery. Here we describe explicitly a leaf-shaped convex region $$\Lambda $$ź contained in $$\varOmega $$Ω and occupying a good portion of $$P$$P. The sets $$\varOmega _T$$ΩT and $$\varOmega _L$$ΩL of translational packing and covering densities and lattice packing and covering densities are defined similarly, restricting the allowed arrangements of $$K$$K to translated copies or lattice arrangements, respectively. Due to affine invariance of the translative and lattice density functions, the sets $$\varOmega _T$$ΩT and $$\varOmega _L$$ΩL are compact. Furthermore, the sets $$\varOmega , \,\varOmega _T$$Ω,ΩT and $$\varOmega _L$$ΩL contain the subsets $$\varOmega ^\star , \,\varOmega _T^\star $$Ωź,ΩTź and $$\varOmega _L^\star $$ΩLź respectively, corresponding to the centrally symmetric convex disks $$K$$K, and our leaf $$\Lambda $$ź is contained in each of $$\varOmega ^\star , \,\varOmega _T^\star $$Ωź,ΩTź and $$\varOmega _L^\star $$ΩLź.