“The Reinhardt Conjecture As an Optimal Control Problem”, 2017-03-03 ():
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.