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. 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 yields a useful radial perturbation exactly when n ≥ 3.   מנחים : פרופ'  מילמן עמנואל ופרופ' יהודיוף אמיר