Convex Hypersurfaces with Hyperplanar Intersections of Their Homothetic Copies

Extending a well-known characteristic property of ellipsoids, we describe all convex solids $K \subset \mathbb{R}^n$, possibly unboun\-ded, with the following property: for any vector $z \in \mathbb{R}^n$ and any scalar $\lambda \ne 0$ such that $K \ne z + \lambda K$, the intersection of the boundaries of $K$ and $z + \lambda K$ lies in a hyperplane. This property is related to hyperplanarity of shadow-boundaries of $K$ and central symmetricity of small 2-dimensional sections of $K$.