
Vietnam
Journal of Mathematics 39:3 (2011) 287307

On the Reinhardt Conjecture

Thomas C. Hales

Department of Mathematics, University of Pittsburgh, PA
15260, USA

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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