Gauss rigidity and volume preservation under preserving curvature deformations for hedgehogs (Q351095)

The author considers Gauss rigidity and Gauss infinitesimal rigidity for hedgehogs of \(\mathbb R^3\) (regarded as Minkowski differences of closed convex surface of \(\mathbb R^3\) with positive Gaussian curvature) and gives an interesting introduction to these topics. Consider a one-parameter family of hedgehogs with a \(\mathbb C^2\) support function, all of them with he same curvature function. The principal result in this work is: Under an appropriate differentiability conditions of this family with respect to the parameter, all the hedgehogs have the same algebraic volume.