The intersection body $IK$ of a star body $K$ in $\mathbb{R}^{n}$ 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 $n \geq 3$ , $I^{2} K = c K$ iff $K$ is a centered ellipsoid, and hence $I K = c K$ 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 $\rho \in L^{\infty }(\mathbb{S}^{n-1})$ satisfies $\operatorname{\mathcal{R}}(\rho ^{n-1}) = c \rho $ for some $c > 0$ iff $\rho $ is constant, where $\operatorname{\mathcal{R}}$ 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 $I K$ , and introduce a continuous version of Steiner symmetrization for Lipschitz star bodies, which (surprisingly) yields a useful radial perturbation exactly when $n\geq 3$ .
