Fixed points of mean section operators (Q6653783)

Let \({\mathcal K}^n\) (\(n\ge 3\)) denote the space of nonempty compact convex sets in \({\mathbb R}^n\), with Minkowski addition and Hausdorff metric. A mapping \(\Phi:{\mathcal K}^n\to{\mathcal K}^n\) is a Minkowski valuation if \(\Phi(K)+\Phi(L) = \Phi(K\cup L)+\Phi(K\cap L)\) whenever \(K,L,K\cup L\in {\mathcal K}^n\). Let \(\mathbf{MVal}_i\) be the set of continuous, translation invariant Minkowski valuations which intertwine rotations and are homogeneous of degree \(i\). By a result of \textit{F. Dorrek} [Geom. Funct. Anal. 27, No. 3, 466--488 (2017; Zbl 1372.52005)] the support function of \(\Phi_i\in \mathbf{MVal}_i\), for \(i\in\{1,\dots,n-1\}\), can be represented by \(h(\Phi_iK,\cdot)= S_i(K,\cdot)*f\), where \(S_i(K,\cdot)\) is the \(i\)th area measure of \(K\), the symbol \(*\) denotes spherical convolution, and \(f\) is an integrable function on the unit sphere \({\mathbb S}^{n-1}\), with additional properties (called the generating function of \(\Phi_i\)). Employing delicate analysis on the sphere, the present authors first find a condition (*) implying that the convolution with \(f\) is a bounded linear operator from \(C({\mathbb S}^{n-1})\) to \(C^2({\mathbb S}^{n-1})\). This enables them to show that for a weakly monotone \(\Phi_i\in \mathbf{MVal}_i\) (\(1<i\le n-1\)), whose generating function satisfies (*), there exists a \(C^2\) neighborhood of the unit ball where the only fixed points of \(\Phi_i^2\) are balls. As an application, they show that for \(2\le j\le n\), there exists a \(C^2\) neighborhood of the unit ball where the only fixed points of \(M_j^2\) are balls (here \(M_j\) is the \(j\)th mean section operator of Goodey and Weil). Under additional assumptions, a similar theorem is proved for \(\Phi_i\in \mathbf{MVal}_i\) (\(1<i\le n-1\)). It is also shown that each weakly monotone \(\Phi_i\in \mathbf{MVal}_i\) (\(1<i\le n-1\)) maps the space of convex bodies with a \(C^2\) support function into itself.