Curvature measures and stability
type: Item
instance of: scholarly article
title: Curvature measures and stability
zbMATH Open document ID: 0981.52003
DOI: 10.1007/BF01221068
author: Peter Kohlmann
published in: Journal of Geometry
publication date: 2002-03-07T00:00:00Z
review text: The article contains a stability result for balls in Euclidean space subject to a pinching condition on one of its curvature measures. Fixing natural numbers \(1\leq r\leq n\) and some \(0< \varepsilon_{0}< {\beta\over 2+\beta}\), where \(\beta:=( 1-({n-r \over n})^{1/ r})^{n+1}\), the author shows that there is a constant \(k\) with the following property: For any convex body \(C\subset {\mathbb R}^{n+1}\), whose curvature measures satisfy \noindent \((1-\varepsilon)\Phi_{n}(C,\cdot)\leq \Phi_{n-r}(C,\cdot) \leq (1+\varepsilon)\Phi_{n}(C,\cdot)\) with \(\varepsilon\leq \varepsilon_{0}\), there is a unit ball \(B_{1}\), such that its Hausdorff distance from \(C\) is bounded by \(d_{H}(C, B_{1})\leq k \varepsilon^{{2 / (n+1)(n+4)}} \;\).
reviewed by: Evangelia Samiou
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 1508002
zbMATH Keywords: curvature measure
zbMATH Keywords: stability
zbMATH Keywords: convex bodies
zbMATH Keywords: pinching
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1508002