{
  "abstract": "We completely characterize the denseness of strongly minimum-attaining Lipschitz functions, a minimum analogue for strongly norm-attaining Lipschitz functions, in terms of bi-Lipschitz embeddings. More precisely, our main result shows that the set of strongly minimum-attaining Lipschitz functions defined on a complete metric space $M$ fails the denseness if and only if $M$ is bi-Lipschitz equivalent to a subset of $\\mathbb{R}$ with positive Lebesgue measure, or equivalently, if $M$ admits a bi-Lipschitz embedding into $\\mathbb{R}$ and $M$ has positive 1-dimensional Hausdorff measure. As a consequence, we provide an isometric characterization of the pure 1-unrectifiability of $M$ in terms of strongly minimum-attaining Lipschitz maps defined on bi-Lipschitz copies of closed subsets of $M$. Several counterexamples showing that the main result cannot be naturally extended to the vector-valued setting are also presented.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11080",
  "author_total": 1,
  "authors": [
    {
      "name": "Geunsu Choi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6c1eqpxxznvah.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11080",
  "fulltext_url": null,
  "id": "6c1eqpxxznvah",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11080",
  "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.FA"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7212265147",
  "pdf_url": "https://arxiv.org/pdf/2609.11080",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6c1eqpxxznvah.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.11080",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11080"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A bi-Lipschitz characterization of strong minimum-attainment for Lipschitz maps",
  "title_text": "A bi-Lipschitz characterization of strong minimum-attainment for Lipschitz maps",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/6c1eqpxxznvah",
  "venue": "arXiv",
  "versions": []
}