On optimal $\lambda$-separable packings in the plane

Abstract

Let $\mathcal{P}$ be a packing of circular disks of radius $\rho>0$ in the Euclidean, spherical, or hyperbolic plane. Let $0\leq\lambda\leq\rho$. We say that $\mathcal{P}$ is a $\lambda$-separable packing of circular disks of radius $\rho$ if the family $\mathcal{P'}$ of disks concentric to the disks of $\mathcal{P}$ having radius $\lambda$ 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 $\rho$ (with $\lambda=0$) and totally separable packings of circular disks of radius $\rho$ (with $\lambda=\rho$). In this note we extend several theorems on the density, tightness, and contact numbers of disk packings and totally separable disk packings to $\lambda$-separable packings of circular disks of radius $\rho$ in the Euclidean, spherical, and hyperbolic plane. In particular, our upper bounds (resp., lower bounds) for the density (resp., tightness) of $\lambda$-separable packings of unit disks in the Euclidean plane are sharp for all $0\leq\lambda\leq 1$ with the extremal values achieved by $\lambda$-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 $0\leq\lambda\leq\rho$ 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 Moln\'ar decomposition, which is obtained from the underlying Delaunay decomposition and as such might be of independent interest.