Optimal cover of a disk with three smaller congruent disks

Abstract

Abstract In 2008 R. Connelly asked how one should place n small disks of radius r to cover the largest possible area of a disk of radius R > r. More specifically, is there always an optimal configuration with n-fold rotational symmetry for small values of n? The answer is known to be positive for n = 2, negative for n = 5, and it has been conjectured to be positive for n = 3 and 4. In this paper, we present a systematic way to list all possible combinatorial structures of optimal configurations, and we prove that for n = 3 there is always an optimal configuration with rotational symmetry of order three.