The ring lemma in three dimensions

Journal article, 2011

Suppose that n cyclically tangent discs with pairwise disjoint interiors are externally tangent to and surround the unit disc. The sharp ring lemma in two dimensions states that no disc has a radius below c (n) (R (2)) = (F (2n-3)-1)(-1)-where F (k) denotes the kth Fibonacci number-and that the lower bound is attained in essentially unique Apollonian configurations. In this article, generalizations of the ring lemma to three dimensions are discussed, a version of the ring lemma in three dimensions is proved, and a natural generalization of the extremal two-dimensional configuration-thought to be extremal in three dimensions-is given. The sharp three-dimensional ring lemma constant of order n is shown to be bounded from below by the two-dimensional constant of order n - 1.