{
  "id": "2011.14508",
  "version": "v1",
  "published": "2020-11-30T02:27:09.000Z",
  "updated": "2020-11-30T02:27:09.000Z",
  "title": "Regularity of the set of points with a non-unique metric projection",
  "authors": [
    "Piotr Hajłasz"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "Erd\\\"os proved in 1946 that if a set $E\\subset\\mathbb{R}^n$ is closed and non-empty, then the set of points in $\\mathbb{R}^n$ with the property that the nearest point in $E$ is not unique, can be covered by countably many surfaces, each of finite $(n-1)$-dimensional measure. We improve the result by obtaining new regularity results for these surfaces in terms of convexity and $C^2$ regularity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-30T02:27:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26B25",
      "28A75",
      "49J52"
    ],
    "keywords": [
      "non-unique metric projection",
      "dimensional measure",
      "nearest point",
      "regularity results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}