Two-point symmetrization and convexity
For a hyperplane \(H\) in \(\mathbb{R}^n\), let \(\sigma_H\) denote the reflection in \(H\), and let \(H^-\), \(H^+\) be the two closed halfspaces bounded by \(H\). The two-point-symmetrization with respect to \(H\) transforms a set \(A\subset \mathbb{R}^n\) into the set \[ ((A\cap \sigma_H A)\cap H^-)\cup((A\cup (\sigma_H A)\cap H^+). \] A similar definition (with great subspheres) is possible in spherical space \(\mathbb{S}^n\). The authors prove (in more general and refined versions) the following result. If \(K\) is an \(n\)-dimensional convex body (in \(\mathbb{R}^n\) or \(\mathbb{S}^n\)) such that every two-point symmetrization transforms \(K\) into a convex set, then \(K\) is a ball. This was conjectured by the reviewer and is used in a paper by \textit{F. Gao, D. Hug} and the reviewer [Math. Notae 41 (2001/02), 159--176 (2003)].
scientific article; zbMATH DE number 2119265