{
  "abstract": "We prove the existence of Steiner minimal trees spanning a finite family of metric spaces in two cases of the Gromov--Hausdorff theory. In the compact Gromov--Hausdorff space $\\mathcal{M}$, every finite family of nonempty compact metric spaces is spanned by a shortest Steiner minimal tree whose Steiner vertices are again nonempty compact metric spaces. In a fixed Gromov--Hausdorff cloud $\\mathcal{C}$, every finite family of proper, isometrically cobounded spaces admits a finite-length Steiner minimal tree all of whose vertices remain in $\\mathcal{C}$. The compact case uses compact replacement through distance profiles and the Hausdorff hyperspace, and the cloud case uses pointed ultralimits.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17400",
  "author_total": 1,
  "authors": [
    {
      "name": "I. N. Mikhailov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/crdc35r8weqmd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17400",
  "fulltext_url": null,
  "id": "crdc35r8weqmd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17400",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7221803295",
  "pdf_url": "https://arxiv.org/pdf/2610.17400",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/crdc35r8weqmd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2610.17400",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17400"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Steiner Minimal Trees in the Gromov--Hausdorff Space and in Gromov--Hausdorff Clouds",
  "title_text": "Steiner Minimal Trees in the Gromov--Hausdorff Space and in Gromov--Hausdorff Clouds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/crdc35r8weqmd",
  "venue": "arXiv",
  "versions": []
}