Covering the plane with copies of a convex disk

We prove that for every convex disk C in the plane there exists a double-lattice covering of the plane with copies of C with density ϑ ≤ 1.2281772. This improves the best previously known upper bound ϑ ≤ 8/(3+2√3) ≈ 1.2376043, due to Kuperberg, but it is still far from the conjectured value ϑ=2π/3√3 ≈ 1.2091993.

[1]  János Pach,et al.  Combinatorial Geometry , 2012 .

[2]  W. Kuperberg.  Covering the Plane with Congruent Copies of a Convex Body , 1989 .

[3]  C. H. Dowker.  On minimum circumscribed polygons , 1944 .

[4]  R. Kershner.  The Number of Circles Covering a Set , 1939 .

[5]  J. Pach,et al.  Combinatorial geometry , 1995, Wiley-Interscience series in discrete mathematics and optimization.

[6]  Ernst Sas,et al.  Über eine Extremumeigenschaft der Ellipsen , 1939 .