{
  "id": "1602.07629",
  "version": "v1",
  "published": "2016-02-24T18:22:27.000Z",
  "updated": "2016-02-24T18:22:27.000Z",
  "title": "A sharp exceptional set estimate for visibility",
  "authors": [
    "Tuomas Orponen"
  ],
  "comment": "6 pages",
  "categories": [
    "math.CA",
    "math.MG"
  ],
  "abstract": "A Borel set $B \\subset \\mathbb{R}^{n}$ is visible from $x \\in \\mathbb{R}^{n}$, if the radial projection of $B$ with base point $x$ has positive $\\mathcal{H}^{n - 1}$ measure. I prove that if $\\dim B > n - 1$, then $B$ is visible from every point $x \\in \\mathbb{R}^{n} \\setminus E$, where $E$ is an exceptional set with dimension $\\dim E \\leq 2(n - 1) - \\dim B$. This is the sharp bound for all $n \\geq 2$. Many parts of the proof were already contained in a recent previous paper by P. Mattila and the author, where a weaker bound for $\\dim E$ was derived as a corollary from a certain slicing theorem. Here, no improvement to the slicing result is obtained; in brief, the main observation of the present paper is that the proof method gives the optimal result, when applied directly to the visibility problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-24T18:22:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sharp exceptional set estimate",
      "radial projection",
      "visibility problem",
      "sharp bound",
      "borel set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160207629O"
    }
  }
}