{
  "id": "1905.02182",
  "version": "v1",
  "published": "2019-05-06T17:56:31.000Z",
  "updated": "2019-05-06T17:56:31.000Z",
  "title": "Leaves decompositions and optimal transport of vector measures",
  "authors": [
    "Krzysztof J. Ciosmak"
  ],
  "comment": "46 pages",
  "categories": [
    "math.MG",
    "math.DG",
    "math.FA"
  ],
  "abstract": "For a given $1$-Lipschitz map $u\\colon\\mathbb{R}^n\\to\\mathbb{R}^m$ we define a partition, up to a set of Lebesgue measure zero, of $\\mathbb{R}^n$ into maximal closed convex sets such that restriction of $u$ is an isometry on this sets. We consider a disintegration of the Lebesgue measure with respect to this partition. We prove that almost every conditional measure associated to a set of dimension $m$ is equivalent to the restriction of the $m$-dimensional Hausdorff measure to its support. We provide a conterexample to the conjecture of Klartag that, given a vector measure on $\\mathbb{R}^n$ with total mass zero, the conditional measures, with respect to certain $1$-Lipschitz map, also have total mass zero. We develop a theory of optimal transport for vector measures and use it to answer the conjecture in the affirmative provided a certain condition is satisfied.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-06T17:56:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A50",
      "49K35",
      "49Q20",
      "51F99",
      "52A20",
      "52A22",
      "52A40",
      "60D05"
    ],
    "keywords": [
      "vector measure",
      "optimal transport",
      "leaves decompositions",
      "total mass zero",
      "lipschitz map"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}