Finite coverings in the hyperbolic plane (Q1764173)

In \(H^2\) (the hyperbolic plane) a hypercycle of associated distance \(r\) is the complete curve whose points are at distance \(r\) from a given line. Let \(C\) be a convex domain with the property that any boundary point of \(C\) is contained in some hypercycle of associated distance \(r\) such that \(C\) lies on the convex side of the curve. If such a \(C\) is covered by a finite number of circular discs of radius \(r\) then the density of the covering is larger than \(2\pi/\sqrt{27}\). This is the main theorem. As a corollary the author shows that if at least two non-overlapping equal circular discs cover a given circular disc in Euclidean or spherical or hyperbolic 2-space then the density of the arrangement is larger than \(2\pi/\sqrt{27}\). (Similar results for \(E^2\) and \(S^2\) were established earlier, see [\textit{L. Fejes Tóth}, Comment. Math. Helv. 23, 342--349 (1949; Zbl 0035.10901)].) This paper completes the solving of problems started decades ago and hence it is important, substantial and interesting.