The Volume of the Convex Hull of a Body and its Homothetic Copies

In this note, we prove the following result. Let K ⊂ ℝ d be a convex body with the origin O in its interior. If there is a number λ ∈ (0, 1) such that the n-dimensional volume of the convex hull of the union of K with the translates of λ K, by a vector x, depends only on the Euclidean norm of x, then K is a Euclidean ball.