{
  "id": "2103.01789",
  "version": "v1",
  "published": "2021-03-02T14:56:01.000Z",
  "updated": "2021-03-02T14:56:01.000Z",
  "title": "Cone and paraboloid points of arbitrary subsets of Euclidean space",
  "authors": [
    "Matthew Hyde",
    "Michele Villa"
  ],
  "comment": "31 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "In this paper we characterise cone points of arbitrary subsets of Euclidean space. Given $E \\subset \\mathbb{R}^n$, $x \\in E$ is a cone point of $E$ if and only if \\begin{align*} \\int_{0}^1 \\beta_{E}^{d,2}(B(x,r))^2 \\frac{dr}{r} < \\infty, \\end{align*} up to a set of zero $d$-measure. The coefficients $\\beta_E^{d,2}$ are a variation of the Jones coefficients. This is a high dimensional counterpart of a theorem of Bishop and Jones from 1994. We also prove similar results for $\\alpha$-paraboloid points, which are the $C^{1,\\alpha}$ rectifiability counterparts to cone points: $x \\in E$ is an $\\alpha$-paraboloid point if and only if \\begin{align*} \\int_0^1 \\frac{\\overline{\\beta}_{E}^{d,2}(B(x,r))^2}{r^{2\\alpha}} \\, \\frac{dr}{r} < \\infty \\end{align*} up to a set of zero $d$-measure. Here, $\\overline{\\beta}^{d,2}_E$ is another variant of the Jones coefficients, introduced by Azzam and Schul.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-02T14:56:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A12",
      "28A75",
      "28A78"
    ],
    "keywords": [
      "paraboloid point",
      "arbitrary subsets",
      "euclidean space",
      "jones coefficients",
      "high dimensional counterpart"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}