{
  "schema_version": 1,
  "problem_number": "AIM-ANALYSIS-0032",
  "title": "Convex Near-Extremizers for the Multidimensional Hermite-Hadamard Inequality",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Larson proved that the boundary-mean Hermite-Hadamard inequality on bounded convex domains in R^n has sharp constant n for nonnegative subharmonic functions. We construct explicit nonnegative convex near-extremizers on collapsing simplices, proving that restricting the test functions to the convex class does not improve this universal constant. The construction uses the positive part of an affine function, with its localization scale larger than the collapsing thickness. We calculate the exact ratio and a quantitative error bound, and show that the same supremum holds for smooth, strictly positive, strictly convex test functions. This resolves the boundary-mean convex subproblem in AIM Shape optimization with surface interactions, Problem 6.1, corresponding to AIM-ANALYSIS-0032 in UnsolvedMath v1.6.0. Neither volume-power normalized sharp constant is determined here; the whole four-constant record remains partial. The all-dimensional proof is supplemented by 4,072 exact arithmetic regression checks. This is an unrefereed preprint; no independent review, formal verification or absolute-priority claim is asserted.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CA",
    "math.MG"
  ],
  "keywords": [
    "AIM-ANALYSIS-0032",
    "Hermite-Hadamard inequality",
    "convex functions",
    "sharp constant",
    "simplex",
    "boundary integral",
    "near-extremizer"
  ],
  "manuscript_version_date": "2026-10-15",
  "publication_date": "2026-10-15",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-15",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-analysis-0032/",
  "pdf_url": "https://eulersolve.org/papers/aim-analysis-0032/paper.pdf?v=378700605e70",
  "doi": "10.5281/zenodo.23403203",
  "zenodo_record_url": "https://zenodo.org/records/23403203",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete proof that the boundary-mean convex sharp constant equals n in every dimension n >= 2. Larson's upper bound is known; the contribution is explicit convex two-scale near-extremizers and their exact ratio. Neither volume-power normalized constant is determined. The whole four-constant frozen AIM-ANALYSIS-0032 record remains partial. Self-audited, unrefereed and AI-assisted; no independent review, formal verification or absolute-priority certification.",
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      "sha256": "378700605e708c71c07888385cbe5b79b9e2c5ca324301bc6b49ce023e05ce6d"
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    "source.zip": {
      "sha256": "77c9529f0a3795b8615baf669d3b55ed1a832b7470942fc276d10ae3ef28d53e"
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    "verification_report.md": {
      "sha256": "86f2a7cc0a8c618474eb188d1fa152be50cd7d91eeda67bd52c6bd377f6feed3"
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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."
}