{
  "id": "1302.2886",
  "version": "v2",
  "published": "2013-02-12T18:40:44.000Z",
  "updated": "2013-09-09T23:09:46.000Z",
  "title": "Symmetric Monge-Kantorovich problems and polar decompositions of vector fields",
  "authors": [
    "Nassif Ghoussoub",
    "Abbas Moameni"
  ],
  "comment": "29 pages, Further updated version - if any - can be downloaded at http://birs.ca/~nassif/",
  "categories": [
    "math.AP"
  ],
  "abstract": "For any given integer $N\\geq 2$, we show that every bounded measurable vector field from a bounded domain $\\Omega$ into $\\R^d$ is $N$-cyclically monotone up to a measure preserving $N$-involution. The proof involves the solution of a multidimensional symmetric Monge-Kantorovich problem, which we first study in the case of a general cost function on a product domain $\\Omega^N$. The polar decomposition described above corresponds to a special cost function derived from the vector field in question (actually $N-1$ of them). In this case, we show that the supremum over all probability measures on $\\Omega^N$ which are invariant under cyclic permutations and with a given first marginal $\\mu$, is attained on a probability measure that is supported on the graph of a function of the form $x\\to (x, Sx, S^2x,..., S^{N-1}x)$, where $S$ is a $\\mu$-measure preserving transformation on $\\Omega$ such that $S^N=I$ a.e. The proof exploits a remarkable duality between such involutions and those Hamiltonians that are $N$-cyclically antisymmetric.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-09-09T23:09:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "vector field",
      "polar decomposition",
      "probability measure",
      "multidimensional symmetric monge-kantorovich problem",
      "general cost function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1302.2886G"
    }
  }
}