{
  "abstract": "A metric space is called all-set-homogeneous if every isometry between two of its subsets extends to an isometry of the whole space. We classify all-set-homogeneous geodesic spaces: besides the classical examples, they include the universal metric trees of finite valence. We also prove that every complete all-set-homogeneous length space is geodesic, and hence the same classification holds in this setting.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26410",
  "author_total": 2,
  "authors": [
    {
      "name": "Nina Lebedeva"
    },
    {
      "name": "Anton Petrunin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/db2fp7rf4334e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26410",
  "fulltext_url": null,
  "id": "db2fp7rf4334e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26410",
  "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.DG"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7214077901",
  "pdf_url": "https://arxiv.org/pdf/2609.26410",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/db2fp7rf4334e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.26410",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26410"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Metric foundations of geometry",
  "title_text": "Metric foundations of geometry",
  "type": "preprint",
  "updated_at": "2026-11-04T15:21:29Z",
  "url": "https://research.draconitic.com/paper/db2fp7rf4334e",
  "venue": "arXiv",
  "versions": []
}