Fixed and periodic points of the intersection body operators of lower orders

Abstract

For the intersection body operator of lower order $I_iK$ of a star body $K$ in $\mathbb{R}^n$, $i\in\{1, 2,\ldots, n-2\}$, we prove that $I_i^2K = cK$ iff $K$ is an origin-symmetric ball, and hence $I_iK = cK$ iff $K$ is an origin-symmetric ball. Combining the recent breakthrough (case $i = n-1$) of Milman, Shabelman and Yehudayoff (Invent. Math., 241 (2025), 509-558), slight modifications of two long-standing questions 8.6 and 8.7 posed by R. Gardner (Page 302, Geometric Tomography, Cambridge University Press, 1995) are completely solved. As applications, we show that for the spherical Radon transform $\mathcal{R}$, a non-negative $\rho\in L^{\infty}(\mathcal{S}^{n-1})$ satisfies $\mathcal{R}(\rho^i) = c\rho$ for some $c>0$ iff $\rho$ is constant. Also, the sharp Busemann intersection type inequalities are established.