{
  "abstract": "Let $X$ and $Y$ be real Banach spaces, and let $\\cC(X)$ and $\\cC(Y)$ denote the families of all nonempty bounded closed convex subsets of $X$ and $Y$, respectively, equipped with the Hausdorff metric. We prove that every surjective isometry $F:\\cC(X)\\to\\cC(Y)$ is induced by a surjective affine isometry of the underlying spaces. More precisely, $F$ maps singleton sets onto singleton sets, and the map $T:X\\to Y$ defined by $F(\\{x\\})=\\{T(x)\\}$ is a surjective affine isometry satisfying \\[ F(A)=T[A]:=\\{T(x):\\;x\\in A\\},\\qquad(A\\in\\cC(X)). \\] In particular, our theorem extends the result of Gruber and Lettl for finite-dimensional Euclidean spaces to arbitrary real Banach spaces, without imposing any additional assumptions on the underlying spaces.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18252",
  "author_total": 4,
  "authors": [
    {
      "name": "Lixin Cheng"
    },
    {
      "name": "Wuyi He"
    },
    {
      "name": "Chulei Liu"
    },
    {
      "name": "Zheming Zheng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0ecty6w5rg0c9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18252",
  "fulltext_url": null,
  "id": "0ecty6w5rg0c9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18252",
  "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": "W7213503917",
  "pdf_url": "https://arxiv.org/pdf/2609.18252",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0ecty6w5rg0c9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.18252",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18252"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Surjective Hausdorff Isometries of Hyperspaces of Bounded Closed Convex Sets",
  "title_text": "Surjective Hausdorff Isometries of Hyperspaces of Bounded Closed Convex Sets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/0ecty6w5rg0c9",
  "venue": "arXiv",
  "versions": []
}