{
  "id": "1205.2515",
  "version": "v1",
  "published": "2012-05-11T13:14:26.000Z",
  "updated": "2012-05-11T13:14:26.000Z",
  "title": "A short proof of Paouris' inequality",
  "authors": [
    "Radosław Adamczak",
    "Rafał Latała",
    "Alexander E. Litvak",
    "Krzysztof Oleszkiewicz",
    "Alain Pajor",
    "Nicole Tomczak-Jaegermann"
  ],
  "categories": [
    "math.PR",
    "math.MG"
  ],
  "abstract": "We give a short proof of a result of G. Paouris on the tail behaviour of the Euclidean norm $|X|$ of an isotropic log-concave random vector $X\\in\\R^n$, stating that for every $t\\geq 1$, $P(|X|\\geq ct\\sqrt n)\\leq \\exp(-t\\sqrt n)$. More precisely we show that for any log-concave random vector $X$ and any $p\\geq 1$, $(E|X|^p)^{1/p}\\sim E |X|+\\sup_{z\\in S^{n-1}}(E |< z,X>|^p)^{1/p}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-05-11T13:14:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B06",
      "46B09",
      "52A23"
    ],
    "keywords": [
      "short proof",
      "inequality",
      "isotropic log-concave random vector",
      "euclidean norm",
      "tail behaviour"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1205.2515A"
    }
  }
}