We consider a symmetrization procedure for convex function in R^n that preserves mixed volumes of the sublevel sets, and for which a Pólya-Szegő type inequality holds. We will obtain a stability improvement for this Pólya-Szegő type inequality, bounding the Pólya-Szegő deficit in terms of the Hausdorff asymmetry index. This result allows us to prove a quantitative version of the Faber-Krahn and Saint-Venant inequalities for the k-Hessian equation, at least in the case when the aforementioned inequalities hold. Then, with similar arguments, we give a quantitative improvement of a comparison result proved by K. Tso for solutions to the k-Hessian equation with Dirichlet boundary condition.