{
  "id": "math/0601638",
  "version": "v1",
  "published": "2006-01-26T12:51:21.000Z",
  "updated": "2006-01-26T12:51:21.000Z",
  "title": "Upper bounds for edge-antipodal and subequilateral polytopes",
  "authors": [
    "Konrad J Swanepoel"
  ],
  "comment": "7 pages",
  "journal": "Periodica Mathematica Hungarica 54 (2007), 99--106.",
  "doi": "10.1007/s-10998-007-1099-0",
  "categories": [
    "math.MG"
  ],
  "abstract": "A polytope in a finite-dimensional normed space is subequilateral if the length in the norm of each of its edges equals its diameter. Subequilateral polytopes occur in the study of two unrelated subjects: surface energy minimizing cones and edge-antipodal polytopes. We show that the number of vertices of a subequilateral polytope in any d-dimensional normed space is bounded above by (d/2+1)^d for any d >= 2. The same upper bound then follows for the number of vertices of the edge-antipodal polytopes introduced by I.Talata (Period. Math. Hungar. 38 (1999), 231--246). This is a constructive improvement to the result of A.P\\'or (to appear) that for each dimension d there exists an upper bound f(d) for the number of vertices of an edge-antipodal d-polytopes. We also show that in d-dimensional Euclidean space the only subequilateral polytopes are equilateral simplices.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-01-26T12:51:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52B12",
      "52B05"
    ],
    "keywords": [
      "upper bound",
      "edge-antipodal polytopes",
      "surface energy minimizing cones",
      "subequilateral polytopes occur",
      "d-dimensional euclidean space"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......1638S"
    }
  }
}