Best Partial Covering of a Convex Domain by Congruent Circles of a Given Total Area

Generalizing results of L. Fejes Toth [3], [5], we prove the following theorem. Let R be a convex domain of area |R| and let S be a finite family of at least two congruent circles of total area t. Then for the area |F| of the part of R covered by the circles of S, the inequality |F|< tf(|R|/t) holds, where f(x) is the area of the intersection of a circle of unit area and a regular hexagon of area x concentric with the circle.