{
  "id": "1607.03279",
  "version": "v1",
  "published": "2016-07-12T09:21:45.000Z",
  "updated": "2016-07-12T09:21:45.000Z",
  "title": "Monotone and convex restrictions of continuous functions",
  "authors": [
    "Zoltan Buczolich"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "Suppose that $f$ belongs to a suitably defined complete metric space $ {{\\cal C}}^{{\\alpha}}$ of H\\\"older $ {\\alpha}$-functions defined on $[0,1]$. We are interested in whether one can find large (in the sense of Hausdorff, or lower/upper Minkowski dimension) sets $A {\\subset} [0,1]$ such that $f|_{A}$ is monotone, or convex/concave. Some of our results are about generic functions in $ {{\\cal C}}^{{\\alpha}}$ like the following one: we prove that for the generic $f\\in C_{1}^{{\\alpha}}[0,1]$, $0< {\\alpha}<2$ for any $A {\\subset} [0,1]$ such that $f|_{A}$ is convex, or concave we have ${\\mathrm{dim}}_{\\mathrm H} A\\leq \\underline{\\mathrm{dim}}_M A\\leq \\max \\{0, {\\alpha}-1 \\}.$ On the other hand, we also have some results about all functions belonging to a certain space. For example the previous result is complemented by the following one: for $1< {\\alpha}\\leq 2$ for any $f\\in C^{{\\alpha}}[0,1]$ there is always a set $A {\\subset}[0,1]$ such that ${\\mathrm{dim}}_{\\mathrm H} A= {\\alpha}-1$ and $f|_{A}$ is convex, or concave on $A$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-07-12T09:21:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26A15",
      "26A12",
      "26A51",
      "28A78"
    ],
    "keywords": [
      "convex restrictions",
      "continuous functions",
      "suitably defined complete metric space",
      "lower/upper minkowski dimension",
      "generic functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}