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Petty's projection conjecture in dimensions at least four

For a convex body K⊂RnK\subset\mathbb R^n the projection body ΠK\Pi K has support function hΠK(u)=voln−1(proju⊥K)h_{\Pi K}(u)=\mathrm{vol}_{n-1}(\mathrm{proj}_{u^\perp}K). The ratio ∣ΠK∣/∣K∣n−1|\Pi K|/|K|^{n-1} is affine invariant. Petty (1971) conjectured that for n≥3n\ge3 it is minimized exactly by ellipsoids, i.e. ∣ΠK∣/∣K∣n−1≥κn−1nκn2−n|\Pi K|/|K|^{n-1}\ge\kappa_{n-1}^n\kappa_n^{2-n} with equality only for ellipsoids. Local versions were proved by Saroglou-Zvavitch and Ivaki, and in August 2026 Chen, Feng, Li, Xi and Xu proved the case n=3n=3. Does Petty's inequality, with its ellipsoid equality case, hold for every convex body in every dimension n≥4n\ge4?

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Argument
Field
Convex geometry; affine isoperimetric inequalities, projection bodies
Posed by
C. M. Petty, Isoperimetric problems, Proc. Conf. Convexity and Combinatorial Geometry, Univ. of Oklahoma (1971)
Year posed
1971
Years open
55y
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
42 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1: for every n≥4n\ge4 and every convex body K⊂RnK\subset\mathbb R^n, ∣ΠK∣/∣K∣n−1≥κn−1nκn2−n|\Pi K|/|K|^{n-1}\ge\kappa_{n-1}^n\kappa_n^{2-n}, with equality iff KK is an ellipsoid; no symmetry or regularity is assumed. The proof bounds a bilinear pairing ∬∣x⋅y∣ dη dζ\iint|x\cdot y|\,d\eta\,d\zeta via a strict spherical-harmonic estimate for norms and a fixed-point choice of affine position. It does not treat n=3n=3, which Chen, Feng, Li, Xi and Xu proved in August 2026; the paper notes that the two together give the full conjecture for n≥3n\ge3. As corollaries (via Lutwak's known implication) it states the Lutwak-Petty lower-degree inequalities for 2≤i≤n−22\le i\le n-2, without equality cases, and a sharp Holmes-Thompson isoperimetric inequality.

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 of the manuscript was read against Petty's conjecture as the paper states it with the 1971 reference; it claims the inequality and the ellipsoid-only equality case for every convex body in every dimension n≥4n\ge4. The proof was not refereed. The Lean challenge ComparatorChallenges/PettyProjectionVolume.lean (listed in lean/formalization.yaml, declaration OAI.PettyProjection.petty_projection_volume) was read here: for n≥4n\ge4 and every compact convex KK with nonempty interior it asserts pettyConstant n≤∣ΠK∣/∣K∣n−1n\le|\Pi K|/|K|^{n-1} and equality iff KK is an affine image of the unit ball, with ΠK\Pi K defined through orthogonal shadow volumes. This states the headline claim. Not rebuilt here. The three-dimensional case is not part of this work and rests on the human preprint of Chen et al.

Sources

Changelog1 change

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