Gauss Rigidity and Volume Preservation Under Preserving Curvature Deformations for Hedgehogs

We consider Gauss rigidity and Gauss infinitesimal rigidity for hedgehogs of ${\mathbb{R}^{3}}$ (regarded as Minkowski differences of closed convex surfaces of ${\mathbb{R}^{3}}$ with positive Gaussian curvature). Besides, we prove under an appropriate differentiability condition that whenever we perform a deformation of a hedgehog so that its curvature function remains constant, its algebraic volume also remains constant.