{
  "id": "1703.01352",
  "version": "v1",
  "published": "2017-03-03T22:28:47.000Z",
  "updated": "2017-03-03T22:28:47.000Z",
  "title": "The Reinhardt Conjecture as an Optimal Control Problem",
  "authors": [
    "Thomas Hales"
  ],
  "comment": "41 pages",
  "categories": [
    "math.OC",
    "math.MG"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-03T22:28:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49J15",
      "34H05",
      "34H05"
    ],
    "keywords": [
      "optimal control problem",
      "reinhardt conjecture",
      "bang-bang control",
      "optimal control theory",
      "smoothed octagon"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}