On the Lattice Packings and Coverings of the Plane with Convex Quadrilaterals

TL;DR

This paper investigates the lattice packing and covering densities of convex quadrilaterals, including all triangles, determines their optimal lattices, and establishes inequalities relating these densities.

Contribution

It extends known results from triangles to all convex quadrilaterals, explicitly computes densities, identifies optimal lattices, and proves new inequalities between these densities.

Findings

Determined lattice packing density for convex quadrilaterals.

Identified lattices that attain these densities.

Proved inequalities relating packing and covering densities.

Abstract

It is well known that the lattice packing density and the lattice covering density of a triangle are $\frac{2}{3}$ and $\frac{3}{2}$ respectively. We also know that the lattices that attain these densities both are unique. Let $\delta_{L}(K)$ and $\vartheta_{L}(K)$ denote the lattice packing density and the lattice covering density of $K$, respectively. In this paper, I study the lattice packings and coverings for a special class of convex disks, which includes all triangles and convex quadrilaterals. In particular, I determine the densities $\delta_{L}(Q)$ and $\vartheta_{L}(Q)$, where $Q$ is an arbitrary convex quadrilateral. Furthermore, I also obtain all of lattices that attain these densities. Finally, I show that $\delta_{L}(Q)\vartheta_{L}(Q)\geq 1$ and $\frac{1}{\delta_{L}(Q)}+\frac{1}{\vartheta_{L}(Q)}\geq 2$, for each convex quadrilateral $Q$.