2013/09/02 by Kuperberg, Włodzimierz
#52C15 #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.1309.0281
For every convex disk K (a convex compact subset of the plane, with non-void interior), the packing density δ(K) and covering density ϑ(K) form an ordered pair of real numbers, \em i.e., a point in \mathbb R2. The set Ω consisting of points assigned this way to all convex disks is the subject of this article. A few known inequalities on δ(K) and ϑ(K) jointly outline a relatively small convex polygon P that contains Ω, while the exact shape of Ω remains a mystery. Here we describe explicitly a leaf-shaped convex region Λ contained in Ω and occupying a good portion of P. The sets ΩT and ΩL 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 ΩT and ΩL are compact. Furthermore, the sets Ω, ΩT and ΩL contain the subsets Ω^⋆, ΩT^⋆ and ΩL^⋆ respectively, corresponding to the centrally symmetric convex disks K, and our leaf Λ is contained in each of Ω^⋆, ΩT^⋆ and ΩL^⋆.