A two-sided estimate for the Gaussian noise stability deficit

• 成果类型: Article
• 署名作者: Eldan, Ronen
• 署名单位: University of Washington; University of Washington Seattle
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-014-0556-6
• 发表日期: 2015
• 页码: 561-624
• 关键词: inequality

摘要

The Gaussian noise-stability of a set is defined by where are standard jointly Gaussian vectors satisfying . Borell's inequality states that for all , among all sets with a given Gaussian measure, the quantity is maximized when is a half-space. We give a novel short proof of this fact, based on stochastic calculus. Moreover, we prove an almost tight, two-sided, dimension-free robustness estimate for this inequality: by introducing a new metric to measure the distance between the set and its corresponding half-space (namely the distance between the two centroids), we show that the deficit can be controlled from both below and above by essentially the same function of the distance, up to logarithmic factors. As a consequence, we also establish the conjectured exponent in the robustness estimate proven by Mossel-Neeman, which uses the total-variation distance as a metric. In the limit , we obtain an improved dimension-free robustness bound for the Gaussian isoperimetric inequality. Our estimates are also valid for a generalized version of stability where more than two correlated vectors are considered.