{
  "id": "1508.06414",
  "version": "v1",
  "published": "2015-08-26T09:00:39.000Z",
  "updated": "2015-08-26T09:00:39.000Z",
  "title": "A bound for the perimeter of inner parallel bodies",
  "authors": [
    "Simon Larson"
  ],
  "comment": "10 pages",
  "categories": [
    "math.MG",
    "math.FA"
  ],
  "abstract": "We provide a sharp lower bound for the perimeter of the inner parallel sets of a convex body $\\Omega$. The bound depends only on the perimeter and inradius of the original body and states that \\[ |\\partial\\Omega_t| \\geq \\Bigl(1-\\frac{t}{R_{in}}\\Bigr)^{n-1}_+ |\\partial \\Omega|. \\] In particular the bound is independent of any regularity properties of $\\partial\\Omega$. As a by-product of the proof we establish precise conditions for equality. The proof, which is rather straightforward, is based on the construction of an extremal set for a certain optimization problem and the use of basic properties of mixed volumes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-08-26T09:00:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52A20",
      "52A38",
      "52A40",
      "52A39"
    ],
    "keywords": [
      "inner parallel bodies",
      "inner parallel sets",
      "sharp lower bound",
      "regularity properties",
      "convex body"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150806414L"
    }
  }
}