{
  "id": "1707.04442",
  "version": "v1",
  "published": "2017-07-14T10:12:03.000Z",
  "updated": "2017-07-14T10:12:03.000Z",
  "title": "On the volume of the John-Löwner ellipsoid",
  "authors": [
    "Grigory Ivanov"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "We find an optimal upper bound on the volume of the John ellipsoid of a $k$-dimensional section of the $n$-dimensional cube, and an optimal lower bound on the volume of the L\\\"owner ellipsoid of a projection of the $n$-dimensional cross-polytope onto a $k$-dimensional subspace. We use these results to give a new proof of Ball's upper bound on the volume of a $k$-dimensional section of the hypercube, and of Barthe's lower bound on the volume of a projection of the $n$-dimensional cross-polytope onto a $k$-dimensional subspace. We settle equality cases in these inequalities. Also, we describe all possible vectors in $\\R^n,$ whose coordinates are the squared lengths of a projection of the standard basis in $\\R^n$ onto a $k$-dimensional subspace.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-14T10:12:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "john-löwner ellipsoid",
      "dimensional subspace",
      "dimensional cross-polytope",
      "dimensional section",
      "optimal lower bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}