{
  "abstract": "Banach asked whether a normed space must be Hilbert if, for one fixed dimension greater than one, all subspaces of that dimension are linearly isometric. We prove the complex case. By the codimension-one reduction, the essential finite-dimensional problem is to characterize a balanced convex body in a complex $(n+1)$-space whose complex hyperplane sections are all complex-linearly equivalent. We show that such a body is a Hermitian ellipsoid. The proof adapts the recent bundle--degree method of Lu and Yang for the real problem, with two features specific to the complex setting. After covariance normalization, write $G=\\operatorname{Aut}{\\mathbb C}(S)$ for the complex-linear symmetry group of a model section $S$. The exact section maps form a principal $G$-bundle over $S^{2n+1}$; reduction to $G^{\\circ}$ places its obstruction in $π{2n}(G^{\\circ})$, which is finite. Pulling back by a suitable positive-degree self-map trivializes this bundle and yields a global Lipschitz family of exact complex-linear section maps. Brouwer degree and a signed degree formula then imply that $p_S^{2n+2}$ and $p_S^{2n+4}$ are real homogeneous polynomials, where $p_S$ is the norm of a model section. Unique factorization forces $p_S^2$ to be quadratic, and phase invariance makes the resulting quadratic form Hermitian. The parallelogram identity then yields the general complex Banach-space statement.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26286",
  "author_total": 5,
  "authors": [
    {
      "name": "Xinan Dai"
    },
    {
      "name": "Wendao Deng"
    },
    {
      "name": "Yidong Shi"
    },
    {
      "name": "Tailin Wu"
    },
    {
      "name": "Yuchen Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/37ma70q9nkg0f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26286",
  "fulltext_url": null,
  "id": "37ma70q9nkg0f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26286",
  "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.FA"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7214078829",
  "pdf_url": "https://arxiv.org/pdf/2609.26286",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/37ma70q9nkg0f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26286",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26286"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The Complex Banach Isometric Conjecture",
  "title_text": "The Complex Banach Isometric Conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/37ma70q9nkg0f",
  "venue": "arXiv",
  "versions": []
}