On the Reinhardt Conjecture

TL;DR

This paper discusses a strategy to prove Reinhardt's conjecture, which identifies the shape with the lowest lattice packing density among centrally symmetric convex domains, proposing that the solution is a smoothed octagon.

Contribution

It formulates the Reinhardt conjecture as a calculus of variations problem and outlines a proof strategy, including conditions for minimizers to be smoothed octagons.

Findings

Reinhardt's problem is an instance of Bolza's calculus of variations.

Minimizers have differentiable boundaries with Lipschitz derivatives.

Piecewise analytic minimizers are smoothed polygons, potentially smoothed octagons.

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.