Differentiability properties of the family of p-parallel bodies

We investigate the differentiability of the quermassintegrals with respect to the one-parameter family of the p-parallel bodies. As in the classical case, we obtain that the volume is always differentiable. Although there is no polynomial expression for a p-sum, the rest of quermassintegrals are differentiable on positive values of the parameter too. We prove a sharp lower bound for the derivative of the support function of the p-inner parallel bodies along with equality conditions.