Fixed and Periodic Points of the Intersection Body Operator

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 $n\ge 3$, $I^{2}K=cK$ 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. To this end, 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 $n\ge 3$.