We consider a symmetrization procedure for convex function in $\mathbb{R}^n$ 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 Hausdor 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.
