{
  "id": "2008.01653",
  "version": "v1",
  "published": "2020-08-04T15:43:38.000Z",
  "updated": "2020-08-04T15:43:38.000Z",
  "title": "Banach-Mazur distances between parallelograms and other affinely regular even-gons",
  "authors": [
    "Marek Lassak"
  ],
  "comment": "2 figures",
  "categories": [
    "math.MG",
    "math.FA"
  ],
  "abstract": "First we explain the positions of the parallelogram with respect to a given centrally symmetric planar convex body $C$ which realize the Banach-Mazur distance to $C$. Next we prove that the Banach-Mazur distance from the parallelogram to the affinely regular hexagon is $\\frac{3}{2}$, showing also all the optimal positions of the parallelogram with respect to the hexagon. Analogously, we deal with the distances to the remaining affinely regular even-gons. Namely, we find the distances to the affinely regular $8j$-gons and $(8j+4)$-gons. Moreover, we estimate and conjecture the distances to the affinely regular $(8j+2)$-gons and $(8j+6)$-gons.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-04T15:43:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52A21"
    ],
    "keywords": [
      "banach-mazur distance",
      "parallelogram",
      "centrally symmetric planar convex body",
      "optimal positions",
      "remaining affinely regular even-gons"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}