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

TL;DR

This paper characterizes fixed points of lower order intersection body operators, proving they are only origin-symmetric balls, and applies these results to spherical Radon transforms and Busemann inequalities.

Contribution

It completely solves two longstanding questions about fixed points of intersection body operators of lower orders, extending recent breakthroughs and applying to Radon transforms and geometric inequalities.

Findings

Fixed points of $I_i$ are origin-symmetric balls.

Characterization of functions satisfying $ ho$-Radon transform equations.

Establishment of sharp Busemann intersection inequalities.

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.