{
  "id": "math/9201206",
  "version": "v1",
  "published": "1989-11-09T15:37:00.000Z",
  "updated": "1989-11-09T15:37:00.000Z",
  "title": "On the volume of the intersection of two $L_p^n$ balls",
  "authors": [
    "Gideon Schechtman",
    "Joel Zinn"
  ],
  "categories": [
    "math.FA",
    "math.MG"
  ],
  "abstract": "This note deals with the following problem, the case $p=1$, $q=2$ of which was introduced to us by Vitali Milman: What is the volume left in the $L_p^n$ ball after removing a t-multiple of the $L_q^n$ ball? Recall that the $L_r^n$ ball is the set $\\{(t_1,t_2,\\dots,t_n);\\ t_i\\in{\\bf R},\\ n^{-1}\\sum_{i=1}^n|t_i|^r\\le 1\\}$ and note that for $0<p<q<\\infty$ the $L_q^n$ ball is contained in the $L_p^n$ ball. In Corollary 4 we show that, after normalizing Lebesgue measure so that the volume of the $L_p^n$ ball is one, the answer to the problem above is of order $e^{-ct^pn^{p/q}}$ for $T<t<{1\\over 2}n^ {{1\\over p}-{1\\over q}}$, where $c$ and $T$ depend on $p$ and $q$ but not on $n$. The main theorem, Theorem 3, deals with the corresponding question for the surface measure of the $L_p^n$ sphere.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1989-11-09T15:37:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "intersection",
      "note deals",
      "vitali milman",
      "volume left",
      "normalizing lebesgue measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}