The simultaneous packing and covering constants in the plane

In 1950, C.A. Rogers introduced and studied two simultaneous packing and covering constants for a convex body and obtained the first general upper bound. Afterwards, these constants have attracted the interests of many authors because, besides their own geometric significance, they are closely related to the packing densities and the covering densities of the convex body, especially to the Minkowski–Hlawka theorem. However, so far our knowledge about them is still very limited. In this paper we will determine the optimal upper bound of the simultaneous packing and covering constants for two-dimensional centrally symmetric convex domains, and characterize the domains attaining this upper bound.