Covering with fat convex discs

According to a theorem of L. Fejes Toth [4], if non-crossing congruent copies of a convex disc K cover a convex hexagon H, then the density of the discs relative to H is at least area K/f(K) (6) where f(K) (6) denotes the maximum area of a hexagon contained in K. We say that a convex disc is r-fat if it is contained in a unit circle C and contains a concentric circle c of radius r. Recently, Heppes [7] showed that the above inequality holds without the non-crossing assumption if K is a 0.8561-fat ellipse. We show that the non-crossing assumption can be omitted if K is an r(o)-fat convex disc with r(o) = 0.933 or an r(1)-fat ellipse with r(1) = 0.741.