• Corpus ID: 117825646

On the Reinhardt Conjecture

  title={On the Reinhardt Conjecture},
  author={Thomas C. Hales},
  journal={arXiv: Metric Geometry},
  • T. Hales
  • Published 23 March 2011
  • Mathematics
  • arXiv: Metric Geometry
In 1934, Reinhardt asked for the centrally symmetric convex domain in the plane whose best lattice packing has the lowest density. He conjectured that the unique solution up to an affine transformation is the smoothed octagon (an octagon rounded at corners by arcs of hyperbolas). This article offers a detailed strategy of proof. In particular, we show that the problem is an instance of the classical problem of Bolza in the calculus of variations. A minimizing solution is known to exist. The… 
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