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

The Set of Packing and Covering Densities of Convex Disks