Best partial covering of a convex domain by congruent circles of a given total area (Q2464360)

A result of L. Fejes Toth states that the density of the part of the plane covered by a system of congruent circles od density \(d\) is at most \(df(1/d)\), where \(f(x)\) is the area of the intersection of a circle of unit area and a regular hexagon of area \(x\) which is concentric with the circle. In the paper under review, the author proves that a following generalization follows: If a convex domain \(R\) and a finite family \(S\) of circles of total area \(t\) in the plane are given, and \(F\) is the part of \(R\) which is covered by \(S\), then \(| F| <tf(| R| /t)\). Here \(| .| \) stands for the area of a point set. The proof uses a formula for \(| F| \) and a lower bounding of the negative amount in this formula.