Convex bodies with equiframed two-dimensional sections
A convex body in \(\mathbb{R}^n\) is called equiframed if all its boundary points belong to the boundary of a parallelepiped of minimal volume containing that convex body (for \(n=2\) this notion is closely related to that of Radon curves). It is well known that convex bodies in \(\mathbb{R}^n\), \(n\geq 3\), centred at the origin and having the property that any of their sections with a 2-subspace is bounded by a Radon curve, are ellipsoids. The author gives a stronger result: Every convex body in \(\mathbb{R}^n\), \(n\geq 3\), which is centred at the origin and has only equiframed intersections with linear 2-subspaces, is necessarily an ellipsoid. Since Radon curves form a proper subfamily of equiframed curves, this is a nice generalization of a well-known result in classical convexity. Applications in real normed linear spaces are given, too.
scientific article; zbMATH DE number 5131220