{
  "schema_version": 1,
  "problem_number": "AIM-CONVEX_GEOMETRY-0001",
  "title": "Minimum Perimeter of Central Hyperplane Sections of the Cube",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For every integer n >= 3, we prove that a central hyperplane section of the side-one real cube has perimeter at least 2(n-1), where perimeter is the (n-2)-dimensional measure of its relative boundary. Equality holds exactly for coordinate sections. The proof is analytic in dimensions three and four and every dimension at least seven; dimensions five and six use complete finite certificates checked with exact arithmetic. Ordered facet densities transfer a cubic negative-moment estimate for section volume to perimeter, while a quantitative comparison of nearby graph sections validates every cell of the finite certificates. This answers the minimum-perimeter subquestion, item 2 of AIM Problem 1, not the other questions in the bundled dataset record.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.MG",
    "math.FA"
  ],
  "keywords": [
    "convex geometry",
    "cube sections",
    "minimum perimeter",
    "computer-assisted proof",
    "AIM-CONVEX_GEOMETRY-0001"
  ],
  "manuscript_version_date": "2026-10-05",
  "publication_date": "2026-10-05",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-05",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-convex_geometry-0001/",
  "pdf_url": "https://eulersolve.org/papers/aim-convex_geometry-0001/paper.pdf?v=b5d656668d6d",
  "doi": "10.5281/zenodo.23154580",
  "zenodo_record_url": "https://zenodo.org/records/23154580",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete proof of the real central-hyperplane minimum-perimeter subquestion (item 2 of AIM Problem 1). The six-part AIM-CONVEX_GEOMETRY-0001 source record remains partially solved: other functionals, codimensions and measures are not settled here. Computer-assisted in dimensions five and six; exact full-cover certificates and checker source are included. Prior facet identities and negative-moment methods are credited. Novelty and absolute priority remain undetermined. AI-assisted, self-audited and unrefereed; no independent peer review or proof-assistant formalization is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "b5d656668d6d49544f65ca901de72d33da8b4caaf0df901b8b23357bda85db80"
    },
    "source.zip": {
      "sha256": "5b2453ea17714caddbf1c5ded0c275c3a6fdc26c8ce209fd195dd2313498bf4f"
    },
    "verification_report.md": {
      "sha256": "f56582ab23c265da705a8453091fc1737d2265078e0c1b501ebae4dd1ff22d18"
    }
  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}