{
  "schema_version": 1,
  "problem_number": "AIM-CONVEX_GEOMETRY-0019",
  "title": "A Geometric-Mean Proof of the Spherical Simplex Volume-Product Bound",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "The product of the normalized solid angles of a simplicial cone in R^n and its positive dual is at most 4^(-n), with equality exactly for orthogonal images of the positive orthant. This note gives a short proof by applying the classical geometric-mean Prekopa-Leindler inequality to two inverse positive definite quadratic forms. The result answers the upper-bound question in Problem 23 of the 2010 AIM workshop list on Mahler's conjecture and duality, corresponding to UnsolvedMath v1.6.0 record AIM-CONVEX_GEOMETRY-0019. The functional inequality and its equality criterion are credited to established literature; the spherical conclusion is a direct application, and no historical priority claim is made. An explicit second variation in unit-diagonal Gram coordinates gives a positive quadratic deficit and a dimension-dependent local stability estimate. The accompanying code supplies symbolic identities and numerical sanity checks, not a formal proof. This is an unrefereed English preprint.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.MG",
    "math.FA"
  ],
  "keywords": [
    "AIM-CONVEX_GEOMETRY-0019",
    "spherical simplex",
    "solid angle",
    "dual cone",
    "Prekopa-Leindler inequality",
    "volume product"
  ],
  "manuscript_version_date": "2026-10-16",
  "publication_date": "2026-10-16",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-16",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-convex-geometry-0019/",
  "pdf_url": "https://eulersolve.org/papers/aim-convex-geometry-0019/paper.pdf?v=adb3d7b14ef7",
  "doi": "10.5281/zenodo.23408241",
  "zenodo_record_url": "https://zenodo.org/records/23408241",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete proof of the spherical-simplex upper bound and equality cases, as a direct application of classical geometric-mean integration. An additional second variation yields dimension-dependent local stability. No general-cone theorem, Euclidean Mahler lower-bound result, new functional inequality or historical priority is claimed. Unrefereed and AI-assisted; no independent review or formal verification is certified.",
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      "sha256": "adb3d7b14ef73c99b8a0aa564664b009701dfbeb62e163dc911bac218525124a"
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    "source.zip": {
      "sha256": "c9db5c5500c16827f6d4c08f26b56c932b5e5e93ff966cd26a1a6420be544c3e"
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    "verification_report.md": {
      "sha256": "6976e85d6a8e8729e3a317b2f3b7d07b729903509d705fc10dc6d6a3781a6ad7"
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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."
}