{
  "id": "1209.5640",
  "version": "v2",
  "published": "2012-09-25T15:15:33.000Z",
  "updated": "2018-06-05T17:26:43.000Z",
  "title": "Partial Regularity for optimal transport maps",
  "authors": [
    "Guido De Philippis",
    "Alessio Figalli"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We prove that, for general cost functions on $\\mathbb{R}^n$, or for the cost $d^2/2$ on a Riemannian manifold, optimal transport maps between smooth densities are always smooth outside a closed singular set of measure zero.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-09-25T15:15:33.000Z",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2018-06-05T17:26:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "optimal transport maps",
      "partial regularity",
      "general cost functions",
      "measure zero",
      "riemannian manifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}