The Generalized Busemann-Petty Problem in Dimensions 2 and 3

Precise 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 $m = 3$.

The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.

What AI did

ChatGPT 5.6 Sol

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.

Provider: OpenAI · Prompt public: unknown · Independence: unknown

Verification boundary

unreviewed

A preprint days old, with no independent review.

Correctness: unknown · statement fidelity: unaudited · peer review: none

Timeline

1. 2026-08-06

   arXiv

   Settles subspace dimensions 2 and 3; the generalized problem stays open for larger m.

Known method families

argument (source-reported)

Source-reported tools: argument.

Independent: unknown · difference confidence: 0

What remains uncertain

VibeMath has not independently audited the mathematical statement, proof, or novelty claim.

• 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.