Quantitative Stability for Anisotropic Gaussian Isoperimetric Inequality

Abstract

We prove a sharp, dimension-free quantitative stability theorem for the Gaussian isoperimetric inequality in general Gauss spaces $(\mathbb{R}^n,γ_A)$, where $A$ is an arbitrary positive-definite matrix. More precisely, we show that the anisotropic Gaussian perimeter deficit controls the square of the anisotropic Gaussian asymmetry with a dimension-free constant, recovering the sharp quadratic stability behavior known in the isotropic setting. The central difficulties are the loss of rotational symmetry and the lack of compactness of the isoperimetric sets. The main novelty of our work is a new symmetrization-based reduction to the small-perturbation regime, together with effective Fuglede-type estimates derived from the weighted constant mean curvature equation. These ingredients overcome the principal obstructions created by anisotropy and extend the proof-from-regularity approach to quantitative isoperimetric stability in the anisotropic setting.