| Title: | On optimal λ-separable packings in the plane |
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| Authors: | ID Bezdek, Károly (Author)
ID Lángi, Zsolt (Author) |
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| Files: |  AMC_Bezdek,Langi_2025.pdf (776,44 KB)
MD5: 4AC36DC392D1D9DBA4B561618159FC01
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| Language: | English |
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| Work type: | Article |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: | ZUP - University of Primorska Press
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| Abstract: | 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. |
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| Keywords: | Euclidean, spherical and hyperbolic plane, λ-separable packing, density, tightness, contact number, refined Molnar decomposition |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 12.03.2025 |
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| Publisher: | Založba Univerze na Primorskem |
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| Year of publishing: | 2025 |
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| Number of pages: | 28 str. |
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| Numbering: | Vol. 25, no. 2, [article no.] P2.04 |
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| PID: | 20.500.12556/RUP-21989  |
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| UDC: | 51 |
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| eISSN: | 1855-3974 |
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| DOI: | https://doi.org/10.26493/1855-3974.3130.d46 |
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| Publication date in RUP: | 21.10.2025 |
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| Views: | 252 |
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| Downloads: | 5 |
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