{
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  "source": "vibemath",
  "problem_id": "vibemathed:generalized-busemann-petty-2-3",
  "title": "The Generalized Busemann-Petty Problem in Dimensions 2 and 3",
  "canonical_statement": "If origin-symmetric convex bodies $K, L \\subset \\mathbb{R}^n$ satisfy $\\mathrm{vol}_m(K \\cap E) \\leq \\mathrm{vol}_m(L \\cap E)$ for every $m$-dimensional subspace $E$ with $1 < m < n$, does $\\mathrm{vol}_n(K) \\leq \\mathrm{vol}_n(L)$ follow? Answered affirmatively for subspace dimensions $m = 2$ and $m = 3$.",
  "plain_summary": "VibeMathed reports this item as partial. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "partial",
  "solution_events": [
    {
      "id": "vibemathed:generalized-busemann-petty-2-3:event",
      "problem_id": "vibemathed:generalized-busemann-petty-2-3",
      "type": "proved",
      "status": "partial",
      "occurred_at": "2026-08-06T00:00:00.000Z",
      "title": "arXiv",
      "summary": "Settles subspace dimensions 2 and 3; the generalized problem stays open for larger m.",
      "ai_contribution": "ai_assisted",
      "attempt_ids": [
        "vibemathed:generalized-busemann-petty-2-3:attempt"
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      "model": "ChatGPT 5.6 Sol",
      "provider": "OpenAI",
      "attempted_at": "2026-08-06T00:00:00.000Z",
      "human_collaborators": [
        "Cheng Lin",
        "Yu-De Liu",
        "Ge Xiong"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The declaration in full: \"Theorem 2.3 was found with the help of ChatGPT 5.6 Sol, which led us to prove Theorem 3.2.\" One named theorem, which unlocked the paper's crucial representation formula.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.05621",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: The Generalized Busemann-Petty Problem in Dimensions 2 and 3",
          "url": "https://vibemathed.com/problem/generalized-busemann-petty-2-3",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
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      "independence": "unknown"
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    {
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      "solution_event_id": "vibemathed:generalized-busemann-petty-2-3:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "A preprint days old, with no independent review.",
      "sources": [
        {
          "label": "VibeMathed: The Generalized Busemann-Petty Problem in Dimensions 2 and 3",
          "url": "https://vibemathed.com/problem/generalized-busemann-petty-2-3",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.05621",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv",
      "url": "https://arxiv.org/abs/2608.05621",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: The Generalized Busemann-Petty Problem in Dimensions 2 and 3",
      "url": "https://vibemathed.com/problem/generalized-busemann-petty-2-3",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "expand",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:generalized-busemann-petty-2-3:signal:recent_partial_result",
      "problem_id": "vibemathed:generalized-busemann-petty-2-3",
      "kind": "recent_partial_result",
      "reason": "A source-reported partial result may support bounded expansion.",
      "generated_at": "2026-09-13T16:28:40.178Z"
    }
  ],
  "uncertainties": [
    "VibeMath has not independently verified the mathematical claim.",
    "AI-attempt independence and training-data exposure are unknown unless explicitly documented.",
    "VibeMath has not independently audited the mathematical statement, proof, or novelty claim."
  ],
  "generated_at": "2026-09-13T16:28:40.178Z"
}