{
  "id": "1201.2342",
  "version": "v3",
  "published": "2012-01-11T16:26:42.000Z",
  "updated": "2013-06-26T15:30:10.000Z",
  "title": "Hessian metrics, CD(K,N)-spaces, and optimal transportation of log-concave measures",
  "authors": [
    "Alexander V. Kolesnikov"
  ],
  "comment": "19 pages, more explanations added",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the optimal transportation mapping $\\nabla \\Phi : \\mathbb{R}^d \\mapsto \\mathbb{R}^d$ pushing forward a probability measure $\\mu = e^{-V} \\ dx$ onto another probability measure $\\nu = e^{-W} \\ dx$. Following a classical approach of E. Calabi we introduce the Riemannian metric $g = D^2 \\Phi$ on $\\mathbb{R}^d$ and study spectral properties of the metric-measure space $M=(\\mathbb{R}^d, g, \\mu)$. We prove, in particular, that $M$ admits a non-negative Bakry--{\\'E}mery tensor provided both $V$ and $W$ are convex. If the target measure $\\nu$ is the Lebesgue measure on a convex set $\\Omega$ and $\\mu$ is log-concave we prove that $M$ is a $CD(K,N)$ space. Applications of these results include some global dimension-free a priori estimates of $\\| D^2 \\Phi\\|$. With the help of comparison techniques on Riemannian manifolds and probabilistic concentration arguments we proof some diameter estimates for $M$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-06-26T15:30:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J60",
      "46E35"
    ],
    "keywords": [
      "optimal transportation",
      "log-concave measures",
      "hessian metrics",
      "probability measure",
      "probabilistic concentration arguments"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1201.2342K"
    }
  }
}