{
  "id": "2406.18428",
  "version": "v1",
  "published": "2024-06-26T15:21:58.000Z",
  "updated": "2024-06-26T15:21:58.000Z",
  "title": "Small volume bodies of constant width with tetrahedral symmetries",
  "authors": [
    "Andrii Arman",
    "Andriy Bondarenko",
    "Andriy Prymak",
    "Danylo Radchenko"
  ],
  "comment": "7 pages, 2 figures",
  "categories": [
    "math.MG"
  ],
  "abstract": "For every $n\\ge 2$, we construct a body $U_n$ of constant width $2$ in $\\mathbb{E}^n$ with small volume and symmetries of a regular $n$-simplex. $U_2$ is the Reuleaux triangle. To the best of our knowledge, $U_3$ was not previously constructed, and its volume is smaller than the volume of other three-dimensional bodies of constant width with tetrahedral symmetries. While the volume of $U_3$ is slightly larger than the volume of Meissner's bodies of width $2$, it exceeds the latter by less than $0.137\\%$. For all large $n$, the volume of $U_n$ is smaller than the volume of the ball of radius $0.891$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-26T15:21:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52A20",
      "52A15",
      "52A23",
      "52A40",
      "28A75",
      "49Q20"
    ],
    "keywords": [
      "constant width",
      "small volume bodies",
      "tetrahedral symmetries",
      "reuleaux triangle",
      "three-dimensional bodies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}