Splitting and pinching for convex sets (Q1912223)

Let \(n\) be a positive integer and let \(l \in \{1, \dots, n\}\). The author shows the existence of a positive number \(\varepsilon (n,l)\) such that for every positive \(\varepsilon < \varepsilon (n,l)\) the following property holds true: if \(C \subset E^{n + 1}\) is an arbitrary closed convex set with nonempty interior and boundary, and with the \((n - l)\)-th curvature measure \(\Phi_{n - l} (C, \cdot)\) being \(\varepsilon\)-close to be boundary measure of \(C\), then \(C\) is congruent to \(E^i \times C'\), where \(0 \leq i \leq n - l\) and where \(C'\) is a convex body in \(E^{n + 1 - i}\). The author conjectures that \(\varepsilon (n,l)\) can be chosen as any positive number smaller than \(l/(2n-l)\). Moreover, the paper presents the related stability theorem for orthogonal disc cylinders. The considerations are based on the Steiner-Schwarz symmetrization processes and generalized Minkowski integral formulas.