Let K be a convex domain. According to the classical result of L. Fejes Tóth [FTL1950], the density of a packing of congruent copies of K in a hexagon cannot be denser than the density of K inside the circumscribed hexagon with minimal area. Besides this statement, we verify that the same density estimate holds for any convex container provided the number of copies is high enough. In addition, we show that if K is a centrally symmetric domain then the inradius and circumradius of the optimal convex container cannot be too different. Following L. Fejes Tóth [FTL1950] in case of coverings, the analogous density estimate is verified under the “noncrossing” assumption, which essentially says that the boundaries of any two congruent copies intersect in two points. In case of both packings and coverings, congruent copies can be replaced by similar copies of not too different sizes. Finally, we verify the hexagon bound for coverings by congruent fat ellipses even without the noncrossing assumption, a result due to A. Heppes.

Concerning the perimeter, we show that the convex domain of minimal perimeter containing n nonoverlapping congruent copies of K gets arbitrarily close to being a circular disc for large n. However, if the perimeter of the compact convex set D covered by n congruent copies of K is maximal then D is close to being a segment for large n.