Expandability radius versus density of a lattice packing (Q1410429)

A set of open domains form a packing if any two of them have no common points. A packing of open circular unit discs is called a lattice packing if the centers of the discs are the points of a lattice. In the present paper lattice packings of open unit discs are considered in the Euclidean plane. The efficiency of a packing is measured by its density, which in case of lattice packings is the quotient of the area of the disc and the area of the fundamental domain of the packing. In this paper, the author presents another way of measuring the efficiency. The notion of the expandability radius \(r_{\exp}\) of a lattice packing of unit discs is introduced and its relation to the density \(d\) of a lattice packing is studied. The expandability radius is the radius of the largest disc which can be used to substitute a disc of the packing without overlapping the rest of the packing and is at least 1. The author shows that for a given expandability radius \(r_{\exp}\) of a lattice packing of unit discs the density \(d\) satisfies \(d\geq\frac{\pi} {2[r_{\exp}+1+\sqrt{(r_{\exp}+1)^2-4}]}\) if \(2\leq r_{\exp}+1\leq \frac 4{\sqrt 3}\), \(d\geq\frac{4\pi} {\sqrt{27}(r_{\exp}+1)^2}\) if \(r_{\exp}+1>\frac 4{\sqrt 3}\), and \(d\leq \max\{d_1(r_{\exp}),d_2 (r_{\exp})\}\), where \(d_1\) and \(d_2\) are determined by the two lattices \(U_1\) and \(U_2\) constructed by the author. The lattices for which the equalities hold are also found. \textit{A. Bezdek} and \textit{W. Kuperberg} [Period. Math. Hung. 34, 3-16 (1997; Zbl 0880.52011)] considered a covering problem. It is said that a covering of the plane with unit circular disks has margin \(\mu\) (\(0\leq\mu\leq 1\)) if any one arbitrarily chosen disk can be replaced with a concentric disk of radius \(r=1-\mu\) and the plane still remains covered. This concept provides a continuous transition from a simple covering (margin 0) to a 2-fold covering (margin 1). They determined the minimum possible density among all lattice coverings with margin \(\mu\). The optimal lattice is the equilateral triangular for \(0 \leq \mu \leq \mu_1\), the square for \(\mu_1 \leq \mu \leq \mu_2\) and the Blundon 2-fold covering for \(\mu_2 \leq \mu \leq 1\), where \(\mu_1=0.56408 \ldots \) and \(\mu_2=0.78408 \ldots \). The present author was able to show similar results for the packing problem discussed above.