{
  "abstract": "Let $(\\mathcal{M},d_{\\mathrm{GH}})$ denote the Gromov-Hausdorff space. We prove that every nonempty compact metric space is isometrically embeddable into $(\\mathcal{M},d_{\\mathrm{GH}})$, answering a question of Iliadis. The corresponding result was previously known for finite metric spaces and several larger classes. However, the general compact case remained open. More precisely, let $D>0$ and let $K$ be a nonempty compact metric space with diameter at most $D$. On a fixed Cantor space, we associate a metric to each $1$-Lipschitz function from $K$ to $[0,D]$ so that the Gromov-Hausdorff distance between any two resulting spaces is exactly the uniform distance between the corresponding functions. The resulting spaces have diameter at most $76D$. As a consequence, the subspace of $\\mathcal{M}$ represented by Cantor spaces is itself isometrically universal for compact metric spaces. We also construct finite approximations of these spaces, obtain an exact formula for the Gromov-Hausdorff distance between the finite models, and derive uniform approximation and covering estimates.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21469",
  "author_total": 1,
  "authors": [
    {
      "name": "Ryo Fukano"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8gjnsm1bbb36e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21469",
  "fulltext_url": null,
  "id": "8gjnsm1bbb36e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21469",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.MG"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.21469",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8gjnsm1bbb36e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2609.21469",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21469"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Every Compact Metric Space Is Isometrically Embeddable into the Gromov-Hausdorff Space",
  "title_text": "Every Compact Metric Space Is Isometrically Embeddable into the Gromov-Hausdorff Space",
  "type": "preprint",
  "updated_at": "2026-11-07T18:20:08Z",
  "url": "https://research.draconitic.com/paper/8gjnsm1bbb36e",
  "venue": "arXiv",
  "versions": []
}