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The Set of Packing and Covering Densities of Convex Disks

Discrete & Computational Geometry, 2013, Vol.50(4), pp.1072-1084 [Peer Reviewed Journal]

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  • Title:
    The Set of Packing and Covering Densities of Convex Disks
  • Author: Kuperberg, Włodzimierz
  • Found In: Discrete & Computational Geometry, 2013, Vol.50(4), pp.1072-1084 [Peer Reviewed Journal]
  • Subjects: Convex disk ; Packing ; Covering ; Density
  • Language: English
  • Description: For every convex disk K (a convex compact subset of the plane, with non-void interior), the packing density delta (K) δ ( K ) and covering density \vartheta (K)} ϑ ( K ) form an ordered pair of real numbers, i.e., a point in mathbb{R }^2 R 2 . 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) δ ( K ) and \vartheta (K)} ϑ ( K ) jointly outline a relatively small convex polygon 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 Λ contained in varOmega Ω and occupying a good portion of P . The sets varOmega _T Ω T and varOmega _L Ω 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 varOmega _T Ω T and varOmega _L Ω L are compact. Furthermore, the sets varOmega , \,\varOmega _T Ω , Ω T and varOmega _L Ω L contain the subsets varOmega ^\star , \,\varOmega _T^\star Ω ⋆ , Ω T ⋆ and varOmega _L^\star Ω L ⋆ respectively, corresponding to the centrally symmetric convex disks K , and our leaf Lambda Λ is contained in each of varOmega ^\star , \,\varOmega _T^\star Ω ⋆ , Ω T ⋆ and varOmega _L^\star Ω L ⋆ .
  • Identifier: ISSN: 0179-5376 ; E-ISSN: 1432-0444 ; DOI: 10.1007/s00454-013-9542-9

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