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 (ℝn,γA), 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.
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BibTeX
@misc{mccurdy2026quantitative,
title = {{Quantitative Stability for Anisotropic Gaussian Isoperimetric Inequality}},
author = {Sean McCurdy and Kuan-Ting Yeh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40876},
eprint = {2610.40876},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40876},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Quantitative Stability for Anisotropic Gaussian Isoperimetric Inequality
AU - Sean McCurdy
AU - Kuan-Ting Yeh
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40876
UR - https://arxiv.org/abs/2610.40876
AB - We prove a sharp, dimension-free quantitative stability theorem for the Gaussian isoperimetric inequality in general Gauss spaces (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.
ER -