{
  "id": "0911.4347",
  "version": "v2",
  "published": "2009-11-23T20:26:28.000Z",
  "updated": "2010-09-06T16:59:19.000Z",
  "title": "A General Duality Theorem for the Monge--Kantorovich Transport Problem",
  "authors": [
    "Mathias Beiglboeck",
    "Christian Leonard",
    "Walter Schachermayer"
  ],
  "categories": [
    "math.OC",
    "math.FA"
  ],
  "abstract": "The duality theory of the Monge--Kantorovich transport problem is analyzed in a general setting. The spaces $X, Y$ are assumed to be polish and equipped with Borel probability measures $\\mu$ and $\\nu$. The transport cost function $c:X\\times Y \\to [0,\\infty]$ is assumed to be Borel. Our main result states that in this setting there is no duality gap, provided the optimal transport problem is formulated in a suitably relaxed way. The relaxed transport problem is defined as the limiting cost of the partial transport of masses $1-\\varepsilon$ from $(X,\\mu)$ to $(Y, \\nu)$, as $\\varepsilon >0$ tends to zero. The classical duality theorems of H.\\ Kellerer, where $c$ is lower semi-continuous or uniformly bounded, quickly follow from these general results.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-09-06T16:59:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49Q20"
    ],
    "keywords": [
      "monge-kantorovich transport problem",
      "general duality theorem",
      "borel probability measures",
      "optimal transport problem",
      "transport cost function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0911.4347B"
    }
  }
}