Convex Hypersurfaces with Hyperplanar Intersections of Their Homothetic Copies

Extending a well-known characteristic property of ellipsoids, we describe all convex solids

, possibly unbounded, with the following property: for any vector

z \in \mathbb{R}^n z ∈ R n

and any scalar

\lambda \ne 0 λ ≠ 0

such that

K \ne z + \lambda K K ≠ z + λ K

, the intersection of the boundaries of

K K

and

z + \lambda K z + λ K

lies in a hyperplane. This property is related to hyperplanarity of shadow-boundaries of

K K

and central symmetricity of small 2-dimensional sections of

K K

.