Fixed Points of Mean Section Operators

Abstract

We characterize rotation equivariant bounded linear operators from C(S n−1 ) to C 2 (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 Φ 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 Φ 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.We transform both integrals to spherical cylinder coordinates. For the left hand side, we haveand for the right hand side,where the final equality is obtained from integration by parts. This yields (3.3).For the second part of the lemma, let ψ ∈ D(−1, 1) be an arbitrary test function. Due to the spherical divergence theorem for 2-tensor fields and (3.2),We transform both integrals to spherical cylinder coordinates. For the left hand side, we have