Covering the plane with fat ellipses without non-crossing assumption
A classical theorem of R. Kershner says that the density of a covering of the Euclidean plane with congruent circular discs cannot be smaller than \(2\pi/\sqrt{27}\). L. Féjes Tóth essentially extended this result to convex centrally symmetric domains. He had given a lower bound for non-crossing copies of a centrally symmetric convex domain covering a convex hexagon of the plane. Two domains are said to cross, if for each of them the part of the domain not belonging to the other domain is not connected. The non-crossing assumption cannot simply be dropped, in general. A suggestion of L. Féjes Tóth was, to investigate this problem with domains which are close to the shape of a circle. Therefore, the author considers fat ellipses, these are ellipses with half axes \(1\) and \(b\), \(1> b\geq b_0=0,8561\dots\), and \(b_0\) can be calculated using the areas of regular \(k\)-gones, inscribed into a unit circle, \(k= 5,6,7\). For a covering of the plane by congruent fad ellipses the author shows the following theorem: If a convex hexagon is covered by a finite number of congruent fat ellipses then the covering density is at least \(2\pi/\sqrt{27}\). For the proof the author uses results of \textit{E. Molnár}, \textit{K. Böröczky} and \textit{G. Féjes Tóth} [Acta Math. Acad. Sci. Hung. 23, 263--270 (1973; Zbl 0252.52006)].
scientific article; zbMATH DE number 1969098