Rigidity and iteration of centroid bodies

This manuscript studies rigidity and global iteration of the classical, volume-normalized centroid body operator. It proves that a convex body homothetic to its centroid body must be a centered ellipsoid in dimensions 3 through 25 and in all sufficiently large dimensions. In the latter range, a stronger result establishes Hausdorff convergence of the ball-normalized iterates of every origin-symmetric convex body to a nondegenerate ellipsoid, without additional rescaling. Results valid in every dimension also force fixed bodies to have a positive-dimensional linear automorphism group and establish ellipsoidal rigidity under a spanning reflection condition, covering linearly unconditional bodies. The proofs combine signed spherical harmonic estimates, dimension-independent regularization, a Gaussian approximation based on two Poisson equations, and nonlinear two-step contraction. Quantitative symmetry restrictions and obstructions to scalar variance estimates and one-step Banach–Mazur contraction complement the main results. The large-dimensional threshold is not explicitly evaluated, and the general classification in the intervening dimensions remains open.