On the ring lemma
The sharp ring lemma states that if n ≥ 3 cyclically tangent discs with pairwise disjoint interiors are externally tangent to and surround the unit disc, then no disc has a radius below cn = (F2n-1 + F2n-2 - 1)-1 – where Fk denotes the kth Fibonacci number – and that the lower bound is attained in essentially unique Apollonian configurations.
Here we give a proof by transforming the problem to a class of strip configurations, after which we closely follow a method of proof due to Aharonov and Stephenson.
Generalizations 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.