{
  "id": "1704.02816",
  "version": "v1",
  "published": "2017-04-10T11:58:52.000Z",
  "updated": "2017-04-10T11:58:52.000Z",
  "title": "Multifractal properties of typical convex functions",
  "authors": [
    "Zoltán Buczolich",
    "Stéphane Seuret"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "We study the singularity (multifractal) spectrum of continuous convex functions defined on $[0,1]^{d}$. Let $E_f({h}) $ be the set of points at which $f$ has a pointwise exponent equal to $h$. We first obtain general upper bounds for the Hausdorff dimension of these sets $E_f(h)$, for all convex functions $f$ and all $h\\geq 0$. We prove that for typical/generic (in the sense of Baire) continuous convex functions $f:[0,1]^{d}\\to \\mathbb{R} $, one has $\\dim E_f(h) =d-2+h$ for all $h\\in[1,2],$ and in addition, we obtain that the set $ E_f({h} )$ is empty if $h\\in (0,1)\\cup (1,+\\infty)$. Also, when $f$ is typical, the boundary of $[0,1]^{d}$ belongs to $E_{f}({0})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-10T11:58:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26B25",
      "28A80"
    ],
    "keywords": [
      "typical convex functions",
      "multifractal properties",
      "continuous convex functions",
      "general upper bounds",
      "hausdorff dimension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}