On a functional equation related to a pair of hedgehogs with congruent projections

Hedgehogs are geometrical objects that describe the Minkowski differences of arbitrary convex bodies in the Euclidean space $\mathbb{E}^n$. We prove that two hedgehogs in $\mathbb{E}^n, n \geq 3$, coincide up to a translation and a reflection in the origin, provided that their projections onto any two-dimensional plane are directly congruent and have no direct rigid motion symmetries. Our result is a consequence of a more general analytic statement about the solutions of a functional equation in which the support functions of hedgehogs are replaced with two arbitrary twice continuously differentiable functions on the unit sphere.