Stability results for convex bodies and related spherical integral transformations
The primary purpose of this article is to prove stability results for intersections of convex bodies with \((d-1)\)-dimensional linear subspaces of \(\mathbb{E}^ d\) and half-spaces. The author introduces the radial power integrals \(P^ d_ \theta\) of a star body \(K\) with radial function \(r_ k\), by integrating the power \(r^ \theta_ k\) (\(\theta\) being any real number) on the unit sphere \(S^{d-1}\). For \(u \in S^{d-1}\), let \(L(u)\) denote the \((d-1)\)- dimensional linear subspace orthogonal to \(u\) and \(L(u)\) the half-space bounded by \(L(u)\) with \(u\) as outer normal vector. Then, assuming \(M\) and \(N\) are convex bodies satisfying certain conditions, he obtains stability estimates such as \[ \begin{aligned} \| P^{d-1}_ \theta (M \cap L(u)) - P^{d-1}_ \theta (N \cap L(u)) \| \leq \varepsilon & \Rightarrow \mu(m,N) \leq f(\varepsilon),\\ {}\| P^{d-1}_ \theta (M\cap \widehat{L}(u)) - P^{d-1}_ \theta (N \cap \widehat{L} (u))\| \leq \varepsilon & \Rightarrow \mu(m,N) \leq g(\varepsilon),\end{aligned} \] where \(\| \cdot \|\) is the \(L_ 2\)-norm on \(C^ 0(S^{d-1})\), \(\mu\) is a metric on the space of convex bodies and \(f\), \(g\) are explicitly known functions depending on \(d\) and on the given conditions for \(M\) and \(N\). The metrics used are the Hausdorff metric, the \(L_ 2\)-metric and analogous metrics defined with regard to the radial function. A section of the paper is devoted to establish relationships between these metrics in order to write the aforementioned estimates in terms of different metrics. In addition, the author obtains corresponding uniqueness statements, which may or not be new, without any additional effort. These results provide nice geometrical interpretations in many particular cases, namely characterizations and stability results for balls, for equiwide or equichordal bodies and characterizations of central symmetry in terms of ``half-volumes'' to name but a few. The relationships with related topics are also indicated in an extensive bibliography. The method employed in the proofs is based on the investigation of stability properties of certain integral transformations for functions on the sphere via the spherical harmonics expansions of these functions.
scientific article; zbMATH DE number 710354