{
  "id": "1908.05650",
  "version": "v1",
  "published": "2019-08-15T17:47:21.000Z",
  "updated": "2019-08-15T17:47:21.000Z",
  "title": "The maximum number of points in the cross-polytope that form a packing set of a scaled cross-polytope",
  "authors": [
    "Ji Hoon Chun"
  ],
  "categories": [
    "math.MG"
  ],
  "abstract": "The problem of finding the largest number of points in the unit cross-polytope such that the $l_{1}$-distance between any two distinct points is at least $2r$ is investigated for $r\\in\\left(1-\\frac{1}{n},1\\right]$ in dimensions $\\geq2$ and for $r\\in\\left(\\frac{1}{2},1\\right]$ in dimension $3$. For the $n$-dimensional cross-polytope, $2n$ points can be placed when $r\\in\\left(1-\\frac{1}{n},1\\right]$. For the three-dimensional cross-polytope, $10$ and $12$ points can be placed if and only if $r\\in\\left(\\frac{3}{5},\\frac{2}{3}\\right]$ and $r\\in\\left(\\frac{4}{7},\\frac{3}{5}\\right]$ respectively, and no more than $14$ points can be placed when $r\\in\\left(\\frac{1}{2},\\frac{4}{7}\\right]$. Also, constructive arrangements of points that attain the upper bounds of $2n$, $10$, and $12$ are provided, as well as $13$ points for dimension $3$ when $r\\in\\left(\\frac{1}{2},\\frac{6}{11}\\right]$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-15T17:47:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "maximum number",
      "packing set",
      "scaled cross-polytope",
      "upper bounds",
      "three-dimensional cross-polytope"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}