Isoperimetric and Geometric Inequalities in Quantitative Form: Stein’s Method Approach

Abstract. We adapt Stein’s method to isoperimetric and geometric inequalities. The main challenge is the treatment of boundary terms. We address this by using an elliptic PDE with an oblique boundary condition. We apply our geometric formulation of Stein’s method to obtain stability in the class of nearly spherical sets for the Brock–Weinstock inequality, the isoperimetric inequality under a constraint on Steklov’s first nonzero eigenvalue, and a combination of weighted and unweighted perimeters. All stability results are formulated with respect to the [Formula: see text]-Zolotarev distance, [Formula: see text], that we introduce to interpolate between the Fraenkel asymmetry and the Kantorovich distance.