We focus on extending the Minkowski problem to hedgehogs. When restricted to the class of convex bodies in \(\mathbb {R}^{n+1}\) whose surface area measures have a density with respect to the spherical Lebesgue measure on \(\mathbb {S}^{n}\) , the classical Minkowski problem can be formulated as the question of existence, uniqueness and regularity of a closed convex hypersurface with a prescribed curvature function. The problem admits a natural extension to hedgehogs; however, for non-convex hedgehogs, it becomes much more difficult—even for \(n=2\) and for \(\mathrm {C}^{\infty }\) -hedgehogs—since it essentially reduces to solving Monge-Ampére equations of mixed type on the unit sphere \(\mathbb {S}^{n}\) . We formulate the uniqueness question and present first partial results. In particular, we consider Gauss rigidity and Gauss infinitesimal rigidity for hedgehogs in \(\mathbb {R}^{3}\) . We do not know if a one-parameter family of \(\mathrm {C}^2\) -hedgehogs, all having the same curvature function, must be congruent in \(\mathbb {R}^{3}\) . Nevertheless, we prove a volume-preservation theorem under curvature-preserving deformations: under an appropriate differentiability condition with respect to the parameter, all hedgehogs in the family have the same algebraic volume.