2014/11/18 by Sriamorn, Kirati
#FOS: Mathematics #Metric Geometry (math.MG) #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1411.4713
It is well known that the lattice packing density and the lattice covering density of a triangle are (2)/(3) and (3)/(2) respectively. We also know that the lattices that attain these densities both are unique. Let δL(K) and ϑL(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 δL(Q) and ϑL(Q), where Q is an arbitrary convex quadrilateral. Furthermore, I also obtain all of lattices that attain these densities. Finally, I show that δL(Q)ϑL(Q)≥ 1 and \frac1δL(Q)+\frac1ϑL(Q)≥ 2, for each convex quadrilateral Q.