{
  "id": "0707.3052",
  "version": "v1",
  "published": "2007-07-20T11:55:33.000Z",
  "updated": "2007-07-20T11:55:33.000Z",
  "title": "Extremal Problems in Minkowski Space related to Minimal Networks",
  "authors": [
    "Konrad J Swanepoel"
  ],
  "comment": "6 pages. 11-year old paper. Implicit question in the last sentence has been answered in Discrete & Computational Geometry 21 (1999) 437-447",
  "journal": "Proceedings of the American Mathematical Society 124 (1996) 2513-2518",
  "doi": "10.1090/S0002-9939-96-03370-9",
  "categories": [
    "math.MG",
    "math.FA"
  ],
  "abstract": "We solve the following problem of Z. F\\\"uredi, J. C. Lagarias and F. Morgan [FLM]: Is there an upper bound polynomial in $n$ for the largest cardinality of a set S of unit vectors in an n-dimensional Minkowski space (or Banach space) such that the sum of any subset has norm less than 1? We prove that |S|\\leq 2n and that equality holds iff the space is linearly isometric to \\ell^n_\\infty, the space with an n-cube as unit ball. We also remark on similar questions raised in [FLM] that arose out of the study of singularities in length-minimizing networks in Minkowski spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-07-20T11:55:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52A40",
      "52A21",
      "49Q10"
    ],
    "keywords": [
      "minimal networks",
      "extremal problems",
      "upper bound polynomial",
      "n-dimensional minkowski space"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "AMS",
      "journal": "Proc. Amer. Math. Soc."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0707.3052S"
    }
  }
}