{
  "id": "1704.08290",
  "version": "v1",
  "published": "2017-04-26T18:44:32.000Z",
  "updated": "2017-04-26T18:44:32.000Z",
  "title": "From $r$-dual sets to uniform contractions",
  "authors": [
    "Karoly Bezdek"
  ],
  "comment": "8 pages",
  "categories": [
    "math.MG"
  ],
  "abstract": "Let $M^d$ denote the $d$-dimensional Euclidean, hyperbolic, or spherical space. The $r$-dual set of given set in $M^d$ is the intersection of balls of radii $r$ centered at the points of the given set. In this paper we prove that for any set of given volume in $M^d$ the volume of the $r$-dual set becomes maximal if the set is a ball. As an application we prove the following. The Kneser-Poulsen Conjecture states that if the centers of a family of $N$ congruent balls in Euclidean $d$-space is contracted, then the volume of the intersection does not decrease. A uniform contraction is a contraction where all the pairwise distances in the first set of centers are larger than all the pairwise distances in the second set of centers. We prove the Kneser-Poulsen conjecture for uniform contractions (with $N$ sufficiently large) in $M^d$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-26T18:44:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "uniform contraction",
      "dual set",
      "kneser-poulsen conjecture states",
      "pairwise distances",
      "dimensional euclidean"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}