Fixed and periodic points of the intersection body operator

The intersection body IK of a star body K in Rn was introduced by E. Lutwak following the work of H. Busemann, and plays a central role in the dual Brunn-Minkowski theory. We show that when n≥3, I2K=cK iff K is a centered ellipsoid, and hence IK=cK iff K is a centered Euclidean ball, answering long-standing questions by Lutwak, Gardner, and Fish–Nazarov–Ryabogin–Zvavitch. An equivalent formulation of the latter in terms of non-linear harmonic analysis states that a non-negative ρ∈L∞(Sn−1) satisfies for some c>0 iff ρ is constant, where denotes the spherical Radon transform. Our proof is entirely geometrical: we recast the iterated intersection body equation as an Euler-Lagrange equation for a certain volume functional under radial perturbations, derive new formulas for the volume of IK, and introduce a continuous version of Steiner symmetrization for Lipschitz star bodies, which (surprisingly) yields a useful radial perturbation exactly when n≥3.