{
  "abstract": "We study the distortion of embedding a union of $k$ Euclidean sets into Euclidean space. We show that a union of $k$ Euclidean sets can be embedded with distortion at most $O(k^{1.128} \\log k)$, independent of the number of points. This is an exponential improvement over the previous best known upper bound of $2^{O(k)}$ obtained by iterating the result of Makarychev and Makarychev (2016) for the special case $k = 2$. Surprisingly, with the additional assumption that each Euclidean set remains Euclidean when we adjoin any other two points, we can obtain an asymptotically tight distortion of $O(\\log k)$, thus, providing a generalization of Bourgain's Theorem (1985). Along the way, we also show that the distortion of the union can be upper bounded by $O(\\log k)$ times the maximum of the Lipschitz extension constants of individual Euclidean sets.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.08546",
  "author_total": 4,
  "authors": [
    {
      "name": "Dimitar Chakarov"
    },
    {
      "name": "Sepideh Mahabadi"
    },
    {
      "name": "Yury Makarychev"
    },
    {
      "name": "Ali Vakilian"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/30grr1gp00aps.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.08546",
  "fulltext_url": null,
  "id": "30grr1gp00aps",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.08546",
  "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",
      "cs.CG"
    ],
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.MG"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.08546",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/30grr1gp00aps.ris",
  "sources": [
    {
      "first_seen": "2026-11-10T06:21:07Z",
      "native_id": "2611.08546",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.08546"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Bourgain's Theorem for a Union of Euclidean Metric Spaces",
  "title_text": "Bourgain's Theorem for a Union of Euclidean Metric Spaces",
  "type": "preprint",
  "updated_at": "2026-11-10T06:21:07Z",
  "url": "https://research.draconitic.com/paper/30grr1gp00aps",
  "venue": "arXiv",
  "versions": []
}