Let $\mathcal{R} : L^\infty(\mathbb{S}^{n-1}) \rightarrow L^\infty(\mathbb{S}^{n-1})$ denote the spherical Radon transform, defined as $\mathcal{R}(f)(\theta) = \int_{\mathbb{S}^{n-1} \cap \theta^{\perp}} f(u) d\sigma(u)$. A long-standing question in non-linear harmonic analysis due to Lutwak, Gardner, and Fish--Nazarov--Ryabogin--Zvavitch, is to characterize those non-negative $\rho \in L^\infty(\mathbb{S}^{n-1})$ so that $\mathcal{R}(\rho^{n-1}) = c \rho$ when $n\geq 3$. We show that this holds iff $\rho$ is constant, and moreover, $\mathcal{R}(\mathcal{R}(\rho^{n-1})^{n-1}) = c \rho$ iff $\rho$ is either identically zero or is the reciprocal of some Euclidean norm. Our proof recasts the problem in a geometric language using the intersection body operator $I$, introduced by Lutwak following the work of Busemann, which plays a central role in the dual Brunn-Minkowski theory. We show that for any star-body $K$ in $\mathbb{R}^n$ 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. To this end, we interpret 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 $n\geq 3$.
Joint work with Shahar Shabelman and Amir Yehudayoff.
