The Reinhardt Conjecture as an Optimal Control Problem
  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.
