Curvature measures and stability
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 \(d_{H}(C, B_{1})\leq k \varepsilon^{{2 / (n+1)(n+4)}} \;\).
scientific article; zbMATH DE number 1508002
scientific article