{
  "abstract": "The Barker--Larman problem asks if a convex body $K \\subseteq \\mathbb R^n$ containing the Euclidean ball $\\mathbb B_n$, such that all the sections of $K$ by hyperplanes tangent to $\\mathbb B_n$ have constant $(n-1)$-dimensional volume, must necessarily be a Euclidean ball. In this paper we show a result pointing to a negative answer in dimension $4$. Taking $λ_0 = 4 \\sqrt{3}π$ and any $N \\in \\mathbb N$, we obtain the existence of a family of convex bodies $K_{λ,N}$ with $λ\\in (λ_0-r_N, λ_0 + r_N)$, such that the sections of $K_{λ,N}$ by hyperplanes tangent to the Euclidean ball, have area within $c |λ- λ_0|^{N+1}$ of $λ$, while the difference between outradius and inradius of $K_{λ,N}$ is larger than $C |λ- λ_0|$. The bodies $K_{λ,N}$ are constructed via radial functions as \\[ρ_{K_{λ,N}}(t) = \\cos\\left( \\sum_{n=0}^N \\frac{(λ-λ_0)^n}{n!} φ_n(t) \\right)^{-1},\\] where $t \\in [0,2π), λ\\in \\mathbb R$ and $φ_n$ are trigonometric polynomials that can be computed explicitly. The convergence of the inner power series when $N \\to \\infty$ (which is left open) would imply a negative answer to the Barker--Larman problem in dimension $4$. As an example we obtain a convex body whose outradius and inradius differ by more than $0.176$, and the area of the sections oscillate by less than $3 \\times 10^{-7}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10257",
  "author_total": 1,
  "authors": [
    {
      "name": "J. Haddad"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ef3pzrc1a614r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10257",
  "fulltext_url": null,
  "id": "ef3pzrc1a614r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10257",
  "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.MG"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7212160923",
  "pdf_url": "https://arxiv.org/pdf/2609.10257",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ef3pzrc1a614r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.10257",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10257"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "An approximate counterexample to the Barker--Larman problem in dimension $4$",
  "title_text": "An approximate counterexample to the Barker--Larman problem in dimension $4$",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/ef3pzrc1a614r",
  "venue": "arXiv",
  "versions": []
}