{
  "id": "2207.13745",
  "version": "v1",
  "published": "2022-07-27T18:49:01.000Z",
  "updated": "2022-07-27T18:49:01.000Z",
  "title": "On regularity of maximal distance minimizers in Euclidean Space",
  "authors": [
    "Alexey Gordeev",
    "Yana Teplitskaya"
  ],
  "comment": "This work is the advanced version of the work arXiv:1910.07630,2019",
  "categories": [
    "math.MG"
  ],
  "abstract": "We study the properties of sets $\\Sigma$ which are the solutions of the maximal distance minimizer problem, i.e. of sets having the minimal length (one-dimensional Hausdorff measure) over the class of closed connected sets $\\Sigma \\subset \\mathbb{R}^n$ satisfying the inequality \\[ max_{y \\in M} dist(y,\\Sigma) \\leq r \\] for a given compact set $M \\subset \\mathbb{R}^n$ and some given $r > 0$. Such sets can be considered as the shortest networks of radiating Wi-Fi cables arriving to each customer (for the set $M$ of customers) at a distance at most $r$. In this paper we prove that any maximal distance minimizer $\\Sigma \\subset \\mathbb{R}^n$ has at most $3$ tangent rays at each point and the angle between any two tangent rays at the same point is at least $2\\pi/3$. Moreover, at the plane (for $n=2$) we show that the number of points with three tangent rays is finite and maximal distance minimizer is a finite union of simple curves with continuous one-sided tangents. All the results are proved for the more general class of local minimizers, i.e. sets which are optimal under a perturbation of a neighbourhood of their arbitrary point.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-27T18:49:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "euclidean space",
      "tangent rays",
      "maximal distance minimizer problem",
      "regularity",
      "one-dimensional hausdorff measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}