# Fixed Points of Mean Section Operators

## TL;DR

This paper characterizes certain rotation-equivariant operators on functions on spheres and applies this to identify fixed points of Minkowski valuations, showing Euclidean balls are unique fixed points under specific conditions.

## Contribution

It provides a new characterization of rotation-equivariant operators via the spherical Laplacian and extends fixed point results for Minkowski valuations.

## Key findings

- Euclidean balls are the only fixed points for certain Minkowski valuations near the unit ball.
- Characterization of operators using the mass distribution of the spherical Laplacian.
- Extension of previous fixed point results by Ivaki (2017) and Schuster (2021).

## Abstract

We characterize rotation equivariant bounded linear operators from $C(\mathbb{S}^{n-1})$ to $C^2(\mathbb{S}^{n-1})$ by the mass distribution of the spherical Laplacian of their kernel function on small polar caps. Using this characterization, we show that every continuous, homogeneous, translation invariant, and rotation equivariant Minkowski valuation $\Phi$ that is weakly monotone maps the space of convex bodies with a $C^2$ support function into itself. As an application, we prove that if $\Phi$ is in addition even or a mean section operator, then Euclidean balls are its only fixed points in some $C^2$ neighborhood of the unit ball. Our approach unifies and extends previous results by Ivaki from 2017 and the second author together with Schuster from 2021.