The ring lemma in three dimensions |
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Authors: | Jonatan Vasilis |
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Affiliation: | 1.Department of Mathematical Sciences,Chalmers University of Technology,Gothenburg,Sweden;2.Department of Mathematical Sciences,University of Gothenburg,Gothenburg,Sweden |
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Abstract: | 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. |
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