Covering with Fat Convex Discs |
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Authors: | Gábor Fejes Tóth |
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Institution: | (1) Rényi Institute of Mathematics, Hungarian Academy of Sciences, Pf. 127, H-1364 Budapest, Hungary |
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Abstract: | 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/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. |
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