{
  "id": "1311.4955",
  "version": "v2",
  "published": "2013-11-20T04:43:56.000Z",
  "updated": "2014-11-22T04:40:19.000Z",
  "title": "On the equivalence between two problems of asymmetry on convex bodies",
  "authors": [
    "Christos Saroglou"
  ],
  "comment": "A serious logical gap fixed, from the previous version",
  "categories": [
    "math.FA"
  ],
  "abstract": "The simplex was conjectured to be the extremal convex body for the two following \"problems of asymmetry\":\\\\ P1) What is the minimal possible value of the quantity $\\max_{K'} |K'|/|K|$? Here, $K'$ ranges over all symmetric convex bodies contained in $K$.\\\\ P2) What is the maximal possible volume of the Blaschke-body of a convex body of volume 1?\\\\ Our main result states that (P1) and (P2) admit precisely the same solutions. This complements a result from [{\\rm K. B\\\"or\\\"oczky, I. B\\'ar\\'any, E. Makai Jr. and J. Pach}, Maximal volume enclosed by plates and proof of the chessboard conjecture], Discrete Math. {\\bf 69} (1986), 101--120], stating that if the simplex solves (P1) then the simplex solves (P2) as well.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-11-20T04:43:56.000Z",
      "abstract": "The simplex was conjectured to be the extremal convex body for the two following ``problems of asymmetry'':\\\\ P1) What is the minimal possible value of the quantity $\\max_{K'} |K'|/|K|$? Here, $K'$ ranges over all symmetric convex bodies contained in $K$.\\\\ P2) What is the maximal possible volume of the Blaschke-body of a convex body of volume 1?\\\\ Our main result states that (P1) and (P2) admit precisely the same solutions. This complements a result from [{\\rm K. B\\\"{o}r\\\"{o}czky, I. B\\'{a}r\\'{a}ny, E. Makai Jr. and J. Pach}, Maximal volume enclosed by plates and proof of the chessboard conjecture], Discrete Math. {\\bf 69} (1986), 101--120], stating that if the simplex solves (P1) then the simplex solves (P2) as well.",
      "comment": "Submitted for publication, 2013",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-11-22T04:40:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "equivalence",
      "main result states",
      "extremal convex body",
      "symmetric convex bodies",
      "makai jr"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.4955S"
    }
  }
}