{
  "id": "1512.06361",
  "version": "v1",
  "published": "2015-12-20T11:51:35.000Z",
  "updated": "2015-12-20T11:51:35.000Z",
  "title": "Sphere covering by minimal number of caps and short closed sets",
  "authors": [
    "A. B. Németh"
  ],
  "categories": [
    "math.GT"
  ],
  "abstract": "A subset of the sphere is said short if it is contained in an open hemisphere. A short closed set which is geodesically convex is called a cap. The following theorem holds: 1. The minimal number of short closed sets covering the $n$-sphere is $n+2$. 2. If $n+2$ short closed sets cover the $n$-sphere then (i) their intersection is empty; (ii) the intersection of any proper subfamily of them is non-empty. In the case of caps (i) and (ii) are also sufficient for the family to be a covering of the sphere.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-20T11:51:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52A45"
    ],
    "keywords": [
      "minimal number",
      "short closed sets cover",
      "sphere covering",
      "theorem holds",
      "open hemisphere"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151206361N"
    }
  }
}