Stability results for convex bodies and related spherical integral transformations
type: Item
instance of: scholarly article
title: Stability results for convex bodies and related spherical integral transformations
zbMATH Open document ID: 0821.52002
published in: Advances in Mathematics
publication date: 1995-09-19T00:00:00Z
review text: 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…
reviewed by: C. Peri
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 44A20
zbMATH DE Number: 710354
zbMATH Keywords: spherical transformations
zbMATH Keywords: stability
zbMATH Keywords: intersections of convex bodies
zbMATH Keywords: subspaces
zbMATH Keywords: half-spaces
zbMATH Keywords: spherical harmonics
author: Helmut Groemer
MaRDI profile type: Publication
full work available at URL: aima.1994.1079
OpenAlex ID: W2028158289
DOI: 10.1006/AIMA.1994.1079
description: scientific article; zbMATH DE number 710354