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Opis: Let P be a packing of circular disks of radius ρ > 0 in the Euclidean, spherical, or hyperbolic plane. Let 0 ≤ λ ≤ ρ. We say that P is a λ-separable packing of circular disks of radius ρ if the family P′ of disks concentric to the disks of P having radius λ form a totally separable packing, i.e., any two disks of P′ can be separated by a line which is disjoint from the interior of every disk of F′. This notion bridges packings of circular disks of radius ρ (with λ = 0) and totally separable packings of circular disks of radius ρ (with λ = ρ). In this note we extend several theorems on the density, tightness, and contact numbers of disk packings and totally separable disk packings to λ-separable packings of circular disks of radius ρ in the Euclidean, spherical, and hyperbolic plane. In particular, our upper bounds (resp., lower bounds) for the density (resp., tightness) of λ-separable packings of unit disks in the Euclidean plane are sharp for all 0 ≤ λ ≤ 1 with the extremal values achieved by λ-separable lattice packings of unit disks. On the other hand, the bounds of similar results in the spherical and hyperbolic planes are not sharp for all 0 ≤ λ ≤ ρ although they do not seem to be far from the relevant optimal bounds either. The proofs use local analytic and elementary geometry and are based on the so-called refined Molnár decomposition, which is obtained from the underlying Delaunay decomposition and as such might be of independent interest.
Ključne besede: Euclidean, spherical and hyperbolic plane, λ-separable packing, density, tightness, contact number, refined Molnar decomposition
Objavljeno v RUP: 21.10.2025; Ogledov: 438; Prenosov: 7
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Objavljeno v RUP: 20.09.2022; Ogledov: 2909; Prenosov: 35
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Ključne besede: (incompletely specified) index generating functions, linear decomposition, linear injective mappings
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Objavljeno v RUP: 30.09.2018; Ogledov: 3918; Prenosov: 31
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Objavljeno v RUP: 09.08.2016; Ogledov: 5120; Prenosov: 37
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