Isoperimetric and geometric inequalities in quantitative form: Stein's method approach

TL;DR

This paper introduces a novel approach using Stein's method with elliptic PDEs to analyze and establish stability in various isoperimetric and geometric inequalities, providing quantitative bounds.

Contribution

It adapts Stein's method to geometric inequalities, addressing boundary term challenges and introducing the $ extalpha$-Zolotarev distance for stability analysis.

Findings

Proves stability of the Brock-Weinstock inequality.

Establishes stability of the isoperimetric inequality under Steklov eigenvalue constraints.

Provides stability results for weighted and unweighted perimeters.

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 of the Brock-Weinstock inequality, stability of the isoperimetric inequality under a constraint on Steklov's first non-zero eigenvalue, and stability for the combination of weighted and unweighted perimeters. All stability results are formulated with respect to the $\alpha$-Zolotarev distance, $\alpha$ $\in$ (0, 1], that we introduce to interpolate between the Fraenkel asymmetry and the Kantorovich distance.