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

Abstract

For the intersection body operator of lower order $I_{i}K$ of a star body $K$ in ${\mathbb{R}}^n$, $i\in \{1,2,\dots ,n-2\}$, we prove that $I_i^{2}K=cK$ iff $K$ is an origin-symmetric ball, and hence $I_{i}K=cK$ 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 $\ddot{I}\in L^{\infty }({\mathcal{S}}^{n-1})$ satisfies $\mathcal{R}(\ddot{I}{}^i)=c\ddot{I}$ for some $c>0$ iff $\ddot{I}$ is constant. Also, the sharp Busemann intersection type inequalities are established.

21 pages and 1 figure