{
  "id": "1705.09253",
  "version": "v1",
  "published": "2017-05-25T16:38:26.000Z",
  "updated": "2017-05-25T16:38:26.000Z",
  "title": "Arrangements of homothets of a convex body II",
  "authors": [
    "Márton Naszódi",
    "Konrad J. Swanepoel"
  ],
  "comment": "9 pages",
  "categories": [
    "math.MG",
    "math.CO"
  ],
  "abstract": "A family of homothets of an o-symmetric convex body K in d-dimensional Euclidean space is called a Minkowski arrangement if no homothet contains the center of any other homothet in its interior. We show that any pairwise intersecting Minkowski arrangement of a d-dimensional convex body has at most $2\\cdot 3^d$ members. This improves a result of Polyanskii (arXiv:1610.04400). Using similar ideas, we also give a proof the following result of Polyanskii: Let $K_1,\\dots,K_n$ be a sequence of homothets of the o-symmetric convex body $K$, such that for any $i<j$, the center of $K_j$ lies on the boundary of $K_i$. Then $n\\leq O(3^d d)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-25T16:38:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52A37"
    ],
    "keywords": [
      "o-symmetric convex body",
      "d-dimensional euclidean space",
      "d-dimensional convex body",
      "polyanskii",
      "homothet contains"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}