{
  "id": "2505.09584",
  "version": "v1",
  "published": "2025-05-14T17:33:55.000Z",
  "updated": "2025-05-14T17:33:55.000Z",
  "title": "Tropical Fermat-Weber Points over Bergman Fans",
  "authors": [
    "Shelby Cox",
    "John Sabol",
    "Roan Talbut",
    "Ruriko Yoshida"
  ],
  "categories": [
    "math.CO",
    "q-bio.PE"
  ],
  "abstract": "Given a finite set $[p]:= \\{1, \\ldots , p\\}$, it is well known that the space of all ultrametrics on $[p]$ is the Bergman fan associated to the matroid underlying the complete graph of the vertex set $[p]$. Lin et al.~showed that the set of tropical Fermat-Weber points might not be contained in the space of ultrametrics on $[p]$ even if all input data points are in the space. Here we consider a more general set up and we focus on Fermat-Weber points with respect to the tropical metric over the Bergman fan associated with a matroid with the ground set of $q$ elements. We show that there always exists a Fermat-Weber point in the Bergman fan for data in the Bergman fan and we describe explicitly the set of all Fermat-Weber points in the Bergman fan. Then we introduce the natural extension of the safety radius introduced by Atteson to the set of Fermat-Weber points in the Bergman fan of a matroid.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-05-14T17:33:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bergman fan",
      "tropical fermat-weber points",
      "input data points",
      "general set",
      "safety radius"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}