Classification of fixed points of the centroid body operator

We classify the fixed points, up to dilation, of the classical volume-normalized centroid body operator in every dimension. We prove that ΓK = cK holds precisely for ellipsoids centered at the origin. The proof combines a signed spherical moment-gap argument in low dimensions with boundary integral operators, compact-group structure, and quantitative regression estimates in higher dimensions. Affine eigenfunctions force a large rotation subspace; curvature and log-concave regression bounds then exclude every proper such subspace. The remaining finite comparisons are certified by exact rational inequalities. The manuscript supplies proofs of the auxiliary analytic estimates and explicitly identifies the external results used.