{
  "id": "1907.02132",
  "version": "v1",
  "published": "2019-07-03T20:57:39.000Z",
  "updated": "2019-07-03T20:57:39.000Z",
  "title": "Metrics on sets of interval partitions with diversity",
  "authors": [
    "Noah Forman",
    "Soumik Pal",
    "Douglas Rizzolo",
    "Matthias Winkel"
  ],
  "comment": "12 pages; this preprint generalises and supersedes the parts of arXiv:1609.06706 concerning topologies on interval partitions",
  "categories": [
    "math.PR"
  ],
  "abstract": "We first consider interval partitions whose complements are Lebesgue-null and introduce a complete metric that induces the same topology as the Hausdorff distance (between complements). This is done using correspondences between intervals. Further restricting to interval partitions with alpha-diversity, we then adjust the metric to incorporate diversities. We show that this second metric space is Lusin. An important feature of this topology is that path-continuity in this topology implies the continuous evolution of diversities. This is important in related work on tree-valued stochastic processes where diversities are branch lengths.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-03T20:57:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "interval partitions",
      "second metric space",
      "tree-valued stochastic processes",
      "complete metric",
      "topology implies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}