Covering with Fat Convex Discs

According to a theorem of L. Fejes Tóth [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 r₀-fat convex disc with r₀ = 0.933 or an r₁-fat ellipse with r₁ = 0.741.

Authors and Affiliations

1. Rényi Institute of Mathematics, Hungarian Academy of Sciences, Pf. 127, H-1364 Budapest, Hungary

   Gábor Fejes Tóth

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Correspondence to Gábor Fejes Tóth.

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