{
  "id": "1809.01463",
  "version": "v1",
  "published": "2018-09-05T12:47:32.000Z",
  "updated": "2018-09-05T12:47:32.000Z",
  "title": "On uniqueness in Steiner problem",
  "authors": [
    "Mikhail Basok",
    "Danila Cherkashin",
    "Nikita Rastegaev",
    "Yana. Teplitskaya"
  ],
  "categories": [
    "math.MG",
    "math.CO"
  ],
  "abstract": "We prove that the set of $n$-point configurations for which solution of the planar Steiner problem is not unique has Hausdorff dimension is at most $2n-1$. Moreover, we show that the Hausdorff dimension of $n$-points configurations on which some locally minimal trees have the same length is also at most $2n-1$. Methods we use essentially requires some analytic structure and some finiteness, so that we prove a similar result for a complete Riemannian analytic manifolds under some apriori assumption on the Steiner problem on them.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-05T12:47:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "uniqueness",
      "hausdorff dimension",
      "complete riemannian analytic manifolds",
      "planar steiner problem",
      "points configurations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}