{
  "id": "2108.07193",
  "version": "v1",
  "published": "2021-08-16T16:03:41.000Z",
  "updated": "2021-08-16T16:03:41.000Z",
  "title": "Leaves decompositions in Euclidean spaces",
  "authors": [
    "Krzysztof J. Ciosmak"
  ],
  "comment": "accepted in Journal de Math\\'ematiques Pures et Appliqu\\'ees; the present preprint is formed from arXiv:1905.02182, which has been split; 28 pages",
  "categories": [
    "math.MG",
    "math.DG",
    "math.FA"
  ],
  "abstract": "We partly extend the localisation technique from convex geometry to the multiple constraints setting. For a given $1$-Lipschitz map $u\\colon\\mathbb{R}^n\\to\\mathbb{R}^m$, $m\\leq n$, we define and prove the existence of a partition of $\\mathbb{R}^n$, up to a set of Lebesgue measure zero, into maximal closed convex sets such that restriction of $u$ is an isometry on these sets. We consider a disintegration, with respect to this partition, of a log-concave measure. We prove that for almost every set of the partition of dimension $m$, the associated conditional measure is log-concave. This result is proven also in the context of the curvature-dimension condition for weighted Riemannian manifolds. This partially confirms a conjecture of Klartag.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-16T16:03:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52A20",
      "52A40",
      "28A50",
      "51F99",
      "52A22",
      "60D05",
      "49Q20"
    ],
    "keywords": [
      "euclidean spaces",
      "leaves decompositions",
      "maximal closed convex sets",
      "lebesgue measure zero",
      "lipschitz map"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}