{
  "id": "1008.1544",
  "version": "v2",
  "published": "2010-08-09T16:48:46.000Z",
  "updated": "2010-08-25T23:40:46.000Z",
  "title": "Regularity of optimal transportation between spaces with different dimensions",
  "authors": [
    "Brendan Pass"
  ],
  "comment": "References updated",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the regularity of solutions to an optimal transportation problem where the dimension of the source is larger than that of the target. We demonstrate that if the target is $c$-convex, then the source has a canonical foliation whose co-dimension is equal to the dimension of the target and the problem reduces to an optimal transportation problem between spaces with equal dimensions. If the $c$-convexity condition fails, we do not expect regularity for arbitrary smooth marginals, but, in the case where the source is 2-dimensional and the target is 1 dimensional, we identify sufficient conditions on the marginals and cost to ensure that the optimal map is continuous.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-08-25T23:40:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "optimal transportation problem",
      "convexity condition fails",
      "arbitrary smooth marginals",
      "optimal map",
      "problem reduces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1008.1544P"
    }
  }
}