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Vietnam
Journal of Mathematics 39:3 (2011) 287-307
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On the Reinhardt Conjecture
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Thomas C. Hales
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Department of Mathematics, University of Pittsburgh, PA
15260, USA
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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.
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by Vietnam Academy of Science and Technology & Vietnam Mathematical
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Published
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