The Reinhardt Conjecture as an Optimal Control Problem

TL;DR

This paper reformulates Reinhardt's conjecture about optimal lattice packings of convex bodies as an optimal control problem, identifying extremal trajectories and reducing it to a finite-dimensional optimization, suggesting a new approach to resolve the conjecture.

Contribution

It introduces a novel optimal control formulation of Reinhardt's conjecture, linking geometric packing problems to Pontryagin extremals and finite-dimensional optimization.

Findings

The smoothed octagon is a Pontryagin extremal with bang-bang control.

The optimal solution to the Reinhardt problem has no singular arcs.

The problem reduces to an optimization on a five-dimensional manifold.

Abstract

In 1934, Reinhardt conjectured that the shape of the centrally symmetric convex body in the plane whose densest lattice packing has the smallest density is a smoothed octagon. This conjecture is still open. We formulate the Reinhardt Conjecture as a problem in optimal control theory. The smoothed octagon is a Pontryagin extremal trajectory with bang-bang control. More generally, the smoothed regular $6k+2$-gon is a Pontryagin extremal with bang-bang control. The smoothed octagon is a strict (micro) local minimum to the optimal control problem. The optimal solution to the Reinhardt problem is a trajectory without singular arcs. The extremal trajectories that do not meet the singular locus have bang-bang controls with finitely many switching times. Finally, we reduce the Reinhardt problem to an optimization problem on a five-dimensional manifold. (Each point on the manifold is an initial condition for a potential Pontryagin extremal lifted trajectory.) We suggest that the Reinhardt conjecture might eventually be fully resolved through optimal control theory. Some proofs are computer-assisted using a computer algebra system.