{
  "id": "2011.10279",
  "version": "v1",
  "published": "2020-11-20T08:50:00.000Z",
  "updated": "2020-11-20T08:50:00.000Z",
  "title": "Lusin-type properties of convex functions and convex bodies",
  "authors": [
    "Daniel Azagra",
    "Piotr Hajłasz"
  ],
  "comment": "15 pages",
  "categories": [
    "math.CA",
    "math.DG"
  ],
  "abstract": "We prove that if $f:\\mathbb{R}^n\\to\\mathbb{R}$ is convex and $A\\subset\\mathbb{R}^n$ has finite measure, then for any $\\varepsilon>0$ there is a convex function $g:\\mathbb{R}^n\\to\\mathbb{R}$ of class $C^{1,1}$ such that $\\mathcal{L}^n(\\{x\\in A:\\, f(x)\\neq g(x)\\})<\\varepsilon$. As an application we deduce that if $W\\subset\\mathbb{R}^n$ is a compact convex body then, for every $\\varepsilon>0$, there exists a convex body $W_{\\varepsilon}$ of class $C^{1,1}$ such that $\\mathcal{H}^{n-1}\\left(\\partial W\\setminus \\partial W_{\\varepsilon}\\right)< \\varepsilon$. We also show that if $f:\\mathbb{R}^n\\to\\mathbb{R}$ is a convex function and $f$ is not of class $C^{1,1}_{\\rm loc}$, then for any $\\varepsilon>0$ there is a convex function $g:\\mathbb{R}^n\\to\\mathbb{R}$ of class $C^{1,1}_{\\rm loc}$ such that $\\mathcal{L}^n(\\{x\\in \\mathbb{R}^n:\\, f(x)\\neq g(x)\\})<\\varepsilon$ if and only if $f$ is essentially coercive, meaning that $\\lim_{|x|\\to\\infty}f(x)-\\ell(x)=\\infty$ for some linear function $\\ell$. A consequence of this result is that, if $S$ is the boundary of some convex set with nonempty interior (not necessarily bounded) in $\\mathbb{R}^n$ and $S$ does not contain any line, then for every $\\varepsilon>0$ there exists a convex hypersurface $S_{\\varepsilon}$ of class $C^{1,1}_{\\textrm{loc}}$ such that $\\mathcal{H}^{n-1}(S\\setminus S_{\\varepsilon})<\\varepsilon$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-20T08:50:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26B25",
      "28A75",
      "41A30",
      "52A20",
      "52A27",
      "53C45"
    ],
    "keywords": [
      "convex function",
      "lusin-type properties",
      "compact convex body",
      "finite measure",
      "linear function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}