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, I² 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. 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 $I K$, and introduce a continuous version of Steiner symmetrization for Lipschitz star-bodies, which (surprisingly) yields a useful radial perturbation exactly when n≥ 3.