On the Covering Densities of Quarter-Convex Disks

It is conjectured that for every convex disk $$K$$K, the translative covering density of $$K$$K and the lattice covering density of $$K$$K 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.