{
  "id": "1501.03907",
  "version": "v1",
  "published": "2015-01-16T08:18:17.000Z",
  "updated": "2015-01-16T08:18:17.000Z",
  "title": "Subdivisions of rotationally symmetric planar convex bodies minimizing the maximum relative diameter",
  "authors": [
    "Antonio Cañete",
    "Uwe Schnell",
    "Salvador Segura Gomis"
  ],
  "categories": [
    "math.MG"
  ],
  "abstract": "In this work we study subdivisions of $k$-rotationally symmetric planar convex bodies that minimize the maximum relative diameter functional. For some particular subdivisions called $k$-partitions, consisting of $k$ curves meeting in an interior vertex, we prove that the so-called \\emph{standard $k$-partition} (given by $k$ equiangular inradius segments) is minimizing for any $k\\in\\mathbb{N}$, $k\\geq 3$. For general subdivisions, we show that the previous result only holds for $k\\leq 6$. We also study the optimal set for this problem, obtaining that for each $k\\in\\mathbb{N}$, $k\\geq 3$, it consists of the intersection of the unit circle with the corresponding regular $k$-gon of certain area. Finally, we also discuss the problem for planar convex sets and large values of $k$, and conjecture the optimal $k$-subdivision in this case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-16T08:18:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rotationally symmetric planar convex bodies",
      "symmetric planar convex bodies minimizing",
      "maximum relative diameter",
      "subdivision"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150103907C"
    }
  }
}