{
  "id": "1810.08400",
  "version": "v1",
  "published": "2018-10-19T08:44:01.000Z",
  "updated": "2018-10-19T08:44:01.000Z",
  "title": "Approximation of rectifiable $1$-currents and weak-$\\ast$ relaxation of the $h$-mass",
  "authors": [
    "Andrea Marchese",
    "Benedikt Wirth"
  ],
  "categories": [
    "math.FA",
    "math.AP",
    "math.OC"
  ],
  "abstract": "Based on Smirnov's decomposition theorem we prove that every rectifiable $1$-current $T$ with finite mass $\\mathbb{M}(T)$ and finite mass $\\mathbb{M}(\\partial T)$ of its boundary $\\partial T$ can be approximated in mass by a sequence of rectifiable $1$-currents $T_n$ with polyhedral boundary $\\partial T_n$ and $\\mathbb{M}(\\partial T_n)$ no larger than $\\mathbb{M}(\\partial T)$. Using this result we can compute the relaxation of the $h$-mass for polyhedral $1$-currents with respect to the joint weak-$\\ast$ convergence of currents and their boundaries. We obtain that this relaxation coincides with the usual $h$-mass for normal currents. This shows that the concepts of so-called generalized branched transport and the $h$-mass are equivalent.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-19T08:44:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "approximation",
      "rectifiable",
      "finite mass",
      "smirnovs decomposition theorem",
      "normal currents"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}