On optimal $λ$-separable packings in the plane
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2023
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866918009584484352 |
|---|---|
| author | Bezdek, Károly Lángi, Zsolt |
| author_facet | Bezdek, Károly Lángi, Zsolt |
| contents | Let $\mathcal{P}$ be a packing of circular disks of radius $ρ>0$ in the Euclidean, spherical, or hyperbolic plane. Let $0\leqλ\leqρ$. We say that $\mathcal{P}$ is a $λ$-separable packing of circular disks of radius $ρ$ if the family $\mathcal{P'}$ of disks concentric to the disks of $\mathcal{P}$ having radius $λ$ form a totally separable packing, i.e., any two disks of $\mathcal{P'}$ can be separated by a line which is disjoint from the interior of every disk of $\mathcal{F'}$. 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 $0\leqλ\leq 1$ 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 $0\leqλ\leqρ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_01575 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On optimal $λ$-separable packings in the plane Bezdek, Károly Lángi, Zsolt Metric Geometry Let $\mathcal{P}$ be a packing of circular disks of radius $ρ>0$ in the Euclidean, spherical, or hyperbolic plane. Let $0\leqλ\leqρ$. We say that $\mathcal{P}$ is a $λ$-separable packing of circular disks of radius $ρ$ if the family $\mathcal{P'}$ of disks concentric to the disks of $\mathcal{P}$ having radius $λ$ form a totally separable packing, i.e., any two disks of $\mathcal{P'}$ can be separated by a line which is disjoint from the interior of every disk of $\mathcal{F'}$. 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 $0\leqλ\leq 1$ 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 $0\leqλ\leqρ$ 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. |
| title | On optimal $λ$-separable packings in the plane |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2305.01575 |