{
  "id": "1603.09162",
  "version": "v1",
  "published": "2016-03-30T12:59:29.000Z",
  "updated": "2016-03-30T12:59:29.000Z",
  "title": "Multifractal properties of convex hulls of typical continuous functions",
  "authors": [
    "Zoltan Buczolich"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "We study the singularity (multifractal) spectrum of the convex hull of the typical/generic continuous functions defined on $[0,1]^{d}$. We denote by ${\\mathbf E}_ { { \\varphi } }^{h} $ the set of points at which $ \\varphi : [0,1]^d\\to {\\mathbb R}$ has a pointwise H\\\"older exponent equal to $h$. Let $H_{f}$ be the convex hull of the graph of $f$, the concave function on the top of $H_{f}$ is denoted by $ { { \\varphi } }_{1,f}( { { \\mathbf x } })=\\max \\{y:( { { \\mathbf x } },y)\\in H_{f} \\}$ and $ { { \\varphi } }_{2,f}( { { \\mathbf x } })=\\min \\{y:( { { \\mathbf x } },y)\\in H_{f} \\}$ denotes the convex function on the bottom of $H_{f}$. We show that there is a dense open subset $ { { \\cal G } } { \\subset } {C[0,1]^d}$ such that for $f\\in { { \\cal G } }$ the following properties are satisfied. For $i=1,2$ the functions $ { { { \\varphi } }_ {i,f}}$ and $f$ coincide only on a set of zero Hausdorff dimension, the functions $ { { { \\varphi } }_ {i,f}}$ are continuously differentiable on $(0,1)^{d}$, ${\\mathbf E}_{ { { \\varphi } }_{i,f}}^{0} $ equals the boundary of $ {[0,1]^d}$, $\\dim_{H}{\\mathbf E}_{ { { \\varphi } }_{i,f}}^{1}=d-1 $, $\\dim_{H}{\\mathbf E}_{ { { \\varphi } }_{i,f}}^{+ { \\infty }}=d $ and ${\\mathbf E}_{ { { \\varphi } }_{i,f}}^{h}= { \\emptyset }$ if $h\\in(0,+ { \\infty }) { \\setminus } \\{1 \\}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-03-30T12:59:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26B25",
      "26B05",
      "28A80"
    ],
    "keywords": [
      "convex hull",
      "typical continuous functions",
      "multifractal properties",
      "dense open subset",
      "zero hausdorff dimension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160309162B"
    }
  }
}