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  "problem_id": "vibemathed:volume-and-projection-inequalities-i-zonoids-and-courtade-s-conjecture",
  "title": "Courtade's conjecture on volumes of Minkowski sums with the ball",
  "canonical_statement": "Although Courtade’s conjecture was originally formulated for general convex bodies, we show that it fails even for zonoids in every dimension at least three.",
  "plain_summary": "VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "resolved",
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    {
      "id": "vibemathed:volume-and-projection-inequalities-i-zonoids-and-courtade-s-conjecture:event",
      "problem_id": "vibemathed:volume-and-projection-inequalities-i-zonoids-and-courtade-s-conjecture",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-08-13T00:00:00.000Z",
      "title": "arXiv",
      "summary": "Courtade conjectured that for convex bodies $B,C\\subset\\mathbb R^n$,\n$$\n(|B||C|)^{1/n}\n+\n(|B_2^n||B_2^n+B+C|)^{1/n}\n\\le\n|B_2^n+B|^{1/n}|B_2^n+C|^{1/n}.\n$$\n\nThe paper proves that this inequality is false in every dimension $n\\ge3$. In dimension $3$, it gives an explicit geometric counterexample using two orthogonal double bodies of revolution. It then strengthens this by constructing zonoid counterexamples in every dimension $n\\ge3$, including a six-generator zonotope in dimension $3$ and smooth perturbative constructions in higher dimensions.",
      "ai_contribution": "ai_assisted",
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      "model": "GPT-5.6 Sol",
      "provider": "OpenAI",
      "attempted_at": "2026-08-13T00:00:00.000Z",
      "human_collaborators": [
        "Matthieu Fradelizi",
        "Alfredo Hubard",
        "Auttawich Manui",
        "Cheikh Saliou Ndiaye",
        "Shouda Wang",
        "Artem Zvavitch"
      ],
      "exposure": "unknown",
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      "outcome": "The authors state that GPT-5.6 Sol was used during preparation of the paper as an auxiliary mathematical tool to explore examples, test determinant computations, and assist with preliminary proof development. The authors state that the final mathematical arguments, statements, and computations are theirs and that they independently verified them.",
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      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.12681",
          "kind": "primary-mathematical-source",
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        {
          "label": "VibeMathed: Courtade's conjecture on volumes of Minkowski sums with the ball",
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      "solution_event_id": "vibemathed:volume-and-projection-inequalities-i-zonoids-and-courtade-s-conjecture:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
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      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
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      "note": "Unreviewed. arXiv 2608.12681 read here; the section \"Acknowledgments and AI assistance disclosure\" states that GPT-5.6 Sol was used as an auxiliary mathematical tool to explore examples, test determinant computations and assist preliminary proof development, and that the authors independently verified all arguments. Author-checked; not refereed; no formalization.",
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          "label": "VibeMathed: Courtade's conjecture on volumes of Minkowski sums with the ball",
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    "VibeMath has not independently verified the mathematical claim.",
    "AI-attempt independence and training-data exposure are unknown unless explicitly documented.",
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  "generated_at": "2026-09-13T16:28:40.178Z"
}