On the Reinhardt Conjecture
Abstract
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 boundary of every minimizer is a differentiable curve with Lipschitz continuous derivative. If a minimizer is piecewise analytic, then it is a smoothed polygon (a polygon rounded at corners by arcs of hyperbolas). To complete the proof of the Reinhardt conjecture, the assumption of piecewise analyticity must be removed, and the conclusion of smoothed polygon must be strengthened to smoothed octagon.
 Publication:

arXiv eprints
 Pub Date:
 March 2011
 arXiv:
 arXiv:1103.4518
 Bibcode:
 2011arXiv1103.4518H
 Keywords:

 Mathematics  Metric Geometry;
 Mathematics  Optimization and Control