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- W2083328260 abstract "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." @default.
- W2083328260 created "2016-06-24" @default.
- W2083328260 creator A5089421531 @default.
- W2083328260 date "2013-09-28" @default.
- W2083328260 modified "2023-09-26" @default.
- W2083328260 title "The Set of Packing and Covering Densities of Convex Disks" @default.
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- W2083328260 doi "https://doi.org/10.1007/s00454-013-9542-9" @default.
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