{
  "id": "1103.2796",
  "version": "v1",
  "published": "2011-03-14T21:47:04.000Z",
  "updated": "2011-03-14T21:47:04.000Z",
  "title": "The Monge Problem for distance cost in geodesic spaces",
  "authors": [
    "Stefano Bianchini",
    "Fabio Cavalletti"
  ],
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish space and dL is a geodesic Borel distance which makes (X,dL) a non branching geodesic space. We show that under the assumption that geodesics are d-continuous and locally compact, we can reduce the transport problem to 1-dimensional transport problems along geodesics. We introduce two assumptions on the transport problem {\\pi} which imply that the conditional probabilities of the first marginal on each geodesic are continuous or absolutely continuous w.r.t. the 1- dimensional Hausdorff distance induced by dL. It is known that this regularity is sufficient for the construction of a transport map. We study also the dynamics of transport along the geodesic, the stability of our conditions and show that in this setting dL-cyclical monotonicity is not sufficient for optimality.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-03-14T21:47:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "monge problem",
      "distance cost",
      "transport problem",
      "geodesic borel distance",
      "non branching geodesic space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1103.2796B"
    }
  }
}