{
  "id": "1608.00858",
  "version": "v1",
  "published": "2016-08-02T15:05:19.000Z",
  "updated": "2016-08-02T15:05:19.000Z",
  "title": "Upper Mikowski dimension estimates for convex restrictions",
  "authors": [
    "Zoltan Buczolich"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "We show that there are functions $f$ in the H\\\"older class $C^{ { \\alpha }}[0,1]$, $1< { \\alpha }<2$ such that $f|_{A}$ is not convex, nor concave for any $A { \\subset } [0,1]$ with $ { \\bar { dim }_M } A> { \\alpha }-1$. Our earlier result shows that for the typical/generic $f\\in { C_ { 1 } ^ { { \\alpha } } [0,1] }$, $0\\leq { \\alpha }<2$ there is always a set $A { \\subset } [0,1]$ such that $f|_A$ is convex and $ { \\bar { dim }_M } A=1$. The analogous statement for monotone restrictions is the following: there are functions $f$ in the H\\\"older class $C^{ { \\alpha }}[0,1]$, $1/2 \\leq { \\alpha }<1$ such that $f|_{A}$ is not monotone on $A { \\subset } [0,1]$ with $ { \\bar { dim }_M } A> { \\alpha }$. This statement is not true for the range of parameters $ { \\alpha }<1/2$ and our theorem for the parameter range $1\\leq { \\alpha } <3/2$ cannot be obtained by integration of the result about monotone restrictions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-02T15:05:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26A15",
      "26A12",
      "26A51",
      "28A78"
    ],
    "keywords": [
      "upper mikowski dimension estimates",
      "convex restrictions",
      "monotone restrictions",
      "earlier result",
      "parameter range"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}