{
  "id": "math/0203275",
  "version": "v3",
  "published": "2002-03-26T23:41:05.000Z",
  "updated": "2002-09-25T19:33:18.000Z",
  "title": "Entropy and the Combinatorial Dimension",
  "authors": [
    "S. Mendelson",
    "R. Vershynin"
  ],
  "comment": "A final version of the paper (Inventiones Math., to appear) Only two applications added: one to Asymptotic Geometry (the optimal Elton Theorem) and the other to empirical processes (the uniform central limit theorem)",
  "doi": "10.1007/s00222-002-0266-3",
  "categories": [
    "math.FA"
  ],
  "abstract": "We solve Talagrand's entropy problem: the L_2-covering numbers of every uniformly bounded class of functions are exponential in its shattering dimension. This extends Dudley's theorem on classes of {0,1}-valued functions, for which the shattering dimension is the Vapnik-Chervonenkis dimension. In convex geometry, the solution means that the entropy of a convex body K is controlled by the maximal dimension of a cube of a fixed side contained in the coordinate projections of K. This has a number of consequences, including the optimal Elton's Theorem and estimates on the uniform central limit theorem in the real valued case.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2002-09-25T19:33:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "combinatorial dimension",
      "uniform central limit theorem",
      "talagrands entropy problem",
      "shattering dimension",
      "extends dudleys theorem"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}