20.500.12556/RUP-21989 On optimal λ-separable packings in the plane Optimalna λ-ločljiva pakiranja v ravnini 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. Euclidean spherical and hyperbolic plane λ-separable packing density tightness contact number refined Molnar decomposition Evklidska sferična in hiperbolična ravnina λ-ločljivo pakiranje gostota tesnost kontaktna številka rafinirana Molnárjeva razstavitev true false true Založba Univerze na Primorskem Angleški jezik Slovenski jezik Članek v reviji 2025-10-21 21:53:28 2025-10-21 21:53:29 2025-11-11 03:06:55 0000-00-00 00:00:00 2025 0 0 28 str. no. 2, [article no.] P2.04 Vol. 25 2025 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 2025-03-12 51 1855-3974 https://doi.org/10.26493/1855-3974.3130.d46 AMC_Bezdek,Langi_2025.pdf AMC_Bezdek,Langi_2025.pdf 1 4AC36DC392D1D9DBA4B561618159FC01 f83d287f64d1a2f86f82501e791d7c90f7a7c1b2f53b9b607bfc3d12033284ea c657c82a-aeb8-11f0-8f0b-005056ac49c0 https://repozitorij.upr.si/Dokument.php?lang=slv&id=31924 Založba Univerze na Primorskem 0 0 0