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

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 ${\delta }_L(Q){\vartheta }_L(Q)\ge 1$ and $\frac{1}{{\delta }_L(Q)}+\frac{1}{{\vartheta }_L(Q)}\ge 2$, for each convex quadrilateral $Q$.

References in corpus (1)

• On the Covering Densities of Quarter-Convex Disks