Petty's projection conjecture in dimensions at least four

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

Exact FrontierDelta

Prior state unknown→proved

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Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Petty's projection conjecture in dimensions at least four · Petty projection conjecture, n >= 4

Confidence: Not scored

Registry verification: lean checked · announcement · partial

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VibeMathed record: Petty's projection conjecture in dimensions at least four

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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 $n\ge4$ . 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 $n\ge4$ and every compact convex $K$ with nonempty interior it asserts pettyConstant $n\le|\Pi K|/|K|^{n-1}$ and equality iff $K$ is an affine image of the unit ball, with $\Pi K$ 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.