{
  "schema_version": 1,
  "problem_number": "AIM-CONVEX_GEOMETRY-0045",
  "title": "Minimum-Volume Central Slabs of the Regular Octahedron",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We determine the minimum volume of the intersection of a regular octahedron with a centered slab of every prescribed width, including all equality directions. For the octahedron with vertices +/-e_i and slab half-width t, the minimizing normal is diagonal below a single threshold tau = 0.4740349423895888... and is a coordinate normal above it, until the slab contains the whole body. The threshold is given exactly by a quadratic over Q(sqrt(3)). The analytic proof derives explicit chamber formulas, reduces the optimization to normals with two equal coordinates, and excludes interior extrema using sign-monotone derivatives and a polynomial resultant. This answers only the three-dimensional minimum-direction part of AIM Fourier-Convex Problem 23(d); higher dimensions and maximum volumes are not addressed.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.MG",
    "math.FA"
  ],
  "keywords": [
    "convex geometry",
    "octahedron",
    "cross-polytope",
    "central slabs",
    "minimum volume",
    "AIM-CONVEX_GEOMETRY-0045"
  ],
  "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-0045/",
  "pdf_url": "https://eulersolve.org/papers/aim-convex_geometry-0045/paper.pdf?v=5f1a159e4044",
  "doi": "10.5281/zenodo.23156791",
  "zenodo_record_url": "https://zenodo.org/records/23156791",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete analytic proof of the three-dimensional minimum-volume central-slab theorem for the regular octahedron, including all equality cases. This answers only the n=3 minimum-direction part of AIM Fourier-Convex Problem 23(d). The original four-part AIM-CONVEX_GEOMETRY-0045 source record is not closed: higher-dimensional extrema, maximum volumes, cube slabs and negative spherical moments remain outside the scope. The known wide-slab minimum and standard simplex formula are credited. The release includes 31 exact symbolic checks, which supplement the written proof and do not constitute proof-assistant formalization. Novelty and absolute priority remain undetermined. AI-assisted, self-audited and unrefereed; no independent peer review is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "5f1a159e40441a651db8090a3b7ff5578973d311d0d21b78c4d22565a068962a"
    },
    "source.zip": {
      "sha256": "597ad412faf50e28251b02b883bf035ba8d3479cc087c3d67c394398e256ca07"
    },
    "verification_report.md": {
      "sha256": "ba75c30cf0fcdb2455597b7240341c04227990a0fd067889de067bfce68462f4"
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  },
  "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."
}