{
  "id": "math/9201203",
  "version": "v1",
  "published": "1989-10-26T14:59:00.000Z",
  "updated": "1989-10-26T14:59:00.000Z",
  "title": "Convex bodies with few faces",
  "authors": [
    "Keith Ball",
    "Alain Pajor"
  ],
  "categories": [
    "math.MG",
    "math.FA"
  ],
  "abstract": "It is proved that if $u_1,\\ldots, u_n$ are vectors in ${\\Bbb R}^k, k\\le n, 1 \\le p < \\infty$ and $$r = ({1\\over k} \\sum ^n_1 |u_i|^p)^{1\\over p}$$ then the volume of the symmetric convex body whose boundary functionals are $\\pm u_1,\\ldots, \\pm u_n$, is bounded from below as $$|\\{ x\\in {\\Bbb R}^k\\colon \\ |\\langle x,u_i\\rangle | \\le 1 \\ \\hbox{for every} \\ i\\}|^{1\\over k} \\ge {1\\over \\sqrt{\\rho}r}.$$ An application to number theory is stated.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1989-10-26T14:59:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52A20",
      "10E05"
    ],
    "keywords": [
      "symmetric convex body",
      "boundary functionals",
      "number theory",
      "application"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}