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.

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.