Covering with Fat Convex Discs

Abstract

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/fK(6) where fK(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 r0-fat convex disc with r0 = 0.933 or an r1-fat ellipse with r1 = 0.741.