On the Covering Densities of Quarter-Convex Disks

Abstract

It is conjectured that for every convex disk , the translative covering density of and the lattice covering density of are identical. It is well known that this conjecture is true for every centrally symmetric convex disk. For the non-symmetric case, we only know that the conjecture is true for triangles (Januszewski in Discrete Comput Geom 43:167-178, 2010). In this paper, we prove the conjecture for a class of convex disks (quarter-convex disks), which includes all triangles and convex quadrilaterals.