{
  "id": "1806.08447",
  "version": "v1",
  "published": "2018-06-21T23:06:38.000Z",
  "updated": "2018-06-21T23:06:38.000Z",
  "title": "Exact computation of the $2+1$ convex hull of a finite set",
  "authors": [
    "Pablo Angulo",
    "Daniel Faraco",
    "Carlos García-Gutiérrez"
  ],
  "comment": "4 figures",
  "categories": [
    "math.AP"
  ],
  "abstract": "We present an algorithm to exactly calculate the $\\mathbb{R}^2\\oplus\\mathbb{R}$-separately convex hull of a finite set of points in $\\mathbb{R}^3$, as introduced in \\cite{KirchheimMullerSverak2003}. When $\\mathbb{R}^3$ is considered as certain subset of $3\\times 2 $ matrices, this algorithm calculates the rank-one convex hull. If $\\mathbb{R}^3$ is identified instead with a subset of $2\\times 3$ matrices, the algorithm is actually calculating the quasiconvex hull, due to a recent result by \\cite{HarrisKirchheimLin18}. The algorithm combines outer approximations based in the locality theorem \\cite[4.7]{Kirchheim2003} with inner approximations to $2+1$ convexity based on \"$(2+1)$-complexes\". The departing point is an outer approximation and by iteratively chopping off \"$D$-prisms\", we prove that an inner approximation to the rank-one convex hull is reached.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-21T23:06:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26B25",
      "52B55",
      "49N99"
    ],
    "keywords": [
      "finite set",
      "exact computation",
      "rank-one convex hull",
      "outer approximation",
      "inner approximation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}