[Paper Review] On the Reinhardt Conjecture

Thomas Hales|arXiv (Cornell University)|Mar 23, 2011

Advanced Mathematical Modeling in Engineering5 references5 citations

TL;DR

This paper formulates a comprehensive strategy to prove the Reinhardt conjecture, which posits that the smoothed octagon minimizes lattice packing density among centrally symmetric convex domains in the plane. By modeling the problem as a Bolza-type calculus of variations problem with convexity constraints, the author establishes that any minimizer must be a $C^1$ curve with Lipschitz derivative, and under piecewise analyticity, must be a smoothed polygon—specifically, the smoothed octagon. The key contribution is reducing the conjecture to a finite-dimensional nonlinear optimization problem over rank-one configurations.

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.

Motivation & Objective

• To prove the Reinhardt conjecture that the smoothed octagon uniquely minimizes lattice packing density among centrally symmetric convex domains in the plane.
• To establish that any minimizer of the packing density functional must be a $C^1$ curve with Lipschitz continuous derivative.
• To show that under the assumption of piecewise analyticity, the minimizer must be a smoothed polygon, i.e., a polygon with hyperbolic arcs replacing corners.
• To reduce the full conjecture to a finite-dimensional nonlinear optimization problem over rank-one configurations, thereby completing the proof if piecewise analyticity is removed.

Proposed method

• Formulates the packing density minimization problem as a classical Bolza problem in the calculus of variations with nonholonomic inequality constraints.
• Represents the boundary of the convex domain using $SL_2(\mathbb{R})$-valued curves $\phi(t)$, with the area integral expressed via $I(\phi) = 3\int_{t_0}^{t_1} (\alpha d\gamma - \gamma d\alpha + \beta d\delta - \delta d\beta)$.
• Imposes convexity constraints via $\sigma_j'(t) \land \sigma_j''(t) \geq 0$ a.e., ensuring the boundary remains convex.
• Applies Noether’s theorem to exploit symmetries of the problem, including invariance under reparametrization and $SL_2(\mathbb{R})$ action.
• Reduces the problem to a first-order system by setting $Y = \phi'$, transforming the convexity constraints into linear inequalities $y_j \land y_j' \geq 0$.
• Proposes a nonlinear optimization problem over a 7-dimensional space (initial state and five links, modulo $SL_2(\mathbb{R})$) to test the conjecture computationally.

Experimental results

Research questions

• RQ1 Is the smoothed octagon the unique minimizer of lattice packing density among centrally symmetric convex domains in the plane, up to affine transformation?
• RQ2 Can the assumption of piecewise analyticity in the minimizer be removed to fully prove the Reinhardt conjecture?
• RQ3 Does every minimizer of the packing density functional have a boundary that is a $C^1$ curve with Lipschitz continuous derivative?
• RQ4 Can the problem be reduced to a finite-dimensional optimization over rank-one configurations, and does this lead to the smoothed octagon as the unique solution?
• RQ5 Is the five-link hyperbolic chain conjecture sufficient to characterize the minimizer, and does it imply the smoothed octagon is the unique solution?

Key findings

• A minimizer of the lattice packing density functional exists and is a $C^1$ curve with Lipschitz continuous derivative.
• Under the assumption of piecewise analyticity, any minimizer must be a smoothed polygon, with boundary composed of finitely many linear segments connected by hyperbolic arcs.
• The smoothed octagon achieves a packing density of $\delta(K) = \frac{8 - \sqrt{32} - \ln 2}{\sqrt{8} - 1} \approx 0.902414$, which is conjectured to be the minimal possible.
• The problem is shown to be an instance of the classical Bolza problem in the calculus of variations with autonomous, fixed-endpoint, and inequality-constrained dynamics.
• The circle satisfies the Euler-Lagrange equations but fails second-order conditions, indicating it is not a minimizer, so the minimizer lies on the boundary of the feasible set.
• The conjecture that a five-link hyperbolic chain with minimal area corresponds to the smoothed octagon, if proven, would complete the proof of the Reinhardt conjecture.