  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.
