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Brannen's simplex conjecture for projection-body volume

For a convex body K⊂RdK\subset\mathbb R^d with projection body ΠK\Pi K (support function hΠK(u)=vold−1(proju⊥K)h_{\Pi K}(u)=\mathrm{vol}_{d-1}(\mathrm{proj}_{u^\perp}K)), the ratio Rd(K)=∣ΠK∣/∣K∣d−1R_d(K)=|\Pi K|/|K|^{d-1} is affine invariant. The simplex has Rd=cd=(d+1)dd/d!R_d=c_d=(d+1)d^d/d!. Brannen (1996) conjectured that simplices maximize RdR_d among all convex bodies, i.e. ∣ΠK∣≤cd∣K∣d−1|\Pi K|\le c_d|K|^{d-1}. Is the simplex the maximizer of normalized projection-body volume in every dimension?

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Construction
Field
Convex geometry; projection bodies, reverse affine isoperimetric inequalities
Posed by
N. S. Brannen, Volumes of projection bodies, Mathematika 43 (1996)
Year posed
1996
Years open
30y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
12 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1: for K=T10×T10⊂R20K=T_{10}\times T_{10}\subset\mathbb R^{20}, R20(K)/c20=121(2010)/(21⋅220)>1R_{20}(K)/c_{20}=121\binom{20}{10}/(21\cdot2^{20})>1, so the simplex bound fails in dimension 20; iterating the product identity Rr+s(A×B)=Rr(A)Rs(B)R_{r+s}(A\times B)=R_r(A)R_s(B) gives Rn(Kn)≥λncnR_n(K_n)\ge\lambda^nc_n for some λ>1\lambda>1 and all large nn. Priority: Feng, Hu, Liu and Xu (preprint, August 2026) already disproved the conjecture in every dimension d≥9d\ge9, and Chen et al. proved it in dimension 3; the paper cites both and presents its result as a direct product construction with an exact value. It does not determine the optimal upper constant or the maximizers.

What the AI did

The release README says all results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1 was read against Brannen's conjecture as the paper states it; it claims R20(T10×T10)/c20=22355476/22020096>1R_{20}(T_{10}\times T_{10})/c_{20}=22355476/22020096>1. The proof (a product identity for projection bodies of polytopes plus an exact simplex computation) was not refereed. The Lean challenges ComparatorChallenges/ProjectionCounterexample.lean (declaration OAI.ProjectionCounterexample.universal_simplex_upper_bound_false, in lean/formalization.yaml) and ComparatorChallenges/ProjectionVolume.lean (OAI.Paper092.product_counterexample) were read here: the first asserts that not every convex body in R20\mathbb R^{20} has ratio at most that of the 20-simplex, the second gives the exact value for the product of two 10-simplices. This states the disproof. Not rebuilt here. The paper's exponential-excess corollary is not formalized. The conjecture had already been refuted for every d≥9d\ge9 by Feng, Hu, Liu and Xu (August 2026), as the paper itself says.

Sources

Changelog1 change

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