{
  "id": "0907.0724",
  "version": "v1",
  "published": "2009-07-03T22:11:16.000Z",
  "updated": "2009-07-03T22:11:16.000Z",
  "title": "Lines, Circles, Planes and Spheres",
  "authors": [
    "George B. Purdy",
    "Justin W. Smith"
  ],
  "comment": "37 pages",
  "journal": "Discrete & Computational Geometry, May 2011, Volume 44, Number 4, pp. 860-882",
  "doi": "10.1007/s00454-010-9270-3",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $S$ be a set of $n$ points in $\\mathbb{R}^3$, no three collinear and not all coplanar. If at most $n-k$ are coplanar and $n$ is sufficiently large, the total number of planes determined is at least $1 + k \\binom{n-k}{2}-\\binom{k}{2}(\\frac{n-k}{2})$. For similar conditions and sufficiently large $n$, (inspired by the work of P. D. T. A. Elliott in \\cite{Ell67}) we also show that the number of spheres determined by $n$ points is at least $1+\\binom{n-1}{3}-t_3^{orchard}(n-1)$, and this bound is best possible under its hypothesis. (By $t_3^{orchard}(n)$, we are denoting the maximum number of three-point lines attainable by a configuration of $n$ points, no four collinear, in the plane, i.e., the classic Orchard Problem.) New lower bounds are also given for both lines and circles.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-07-03T22:11:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "51D20"
    ],
    "keywords": [
      "sufficiently large",
      "classic orchard problem",
      "lower bounds",
      "total number",
      "similar conditions"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0907.0724P"
    }
  }
}