{
  "id": "math/0110260",
  "version": "v1",
  "published": "2001-10-23T22:16:55.000Z",
  "updated": "2001-10-23T22:16:55.000Z",
  "title": "On the existence of completely saturated packings and completely reduced covering",
  "authors": [
    "Lewis Bowen"
  ],
  "comment": "14 pages, 1 figure",
  "categories": [
    "math.MG"
  ],
  "abstract": "A packing by a body $K$ is collection of congruent copies of $K$ (in either Euclidean or hyperbolic space) so that no two copies intersect nontrivially in their interiors. A covering by $K$ is a collection of congruent copies of $K$ such that for every point $p$ in the space there is copy in the collection containing $p$. A completely saturated packing is one in which it is not possible to replace a finite number of bodies of the packing with a larger number and still remain a packing. A completely reduced covering is one in which it is not possible to replace a finite number of bodies of the covering with a smaller number and still remain a covering. It was conjectured by G. Fejes Toth, G. Kuperberg, and W. Kuperberg that completely saturated packings and commpletely reduced coverings exist for every body $K$ in either $n$-dimensional Euclidean or $n$-dimensional hyperbolic space. We prove this conjecture.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2001-10-23T22:16:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52C17",
      "52A40",
      "52C26"
    ],
    "keywords": [
      "saturated packing",
      "reduced covering",
      "finite number",
      "congruent copies",
      "dimensional hyperbolic space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2001math.....10260B"
    }
  }
}