ROBUST DIMENSION FREE ISOPERIMETRY IN GAUSSIAN SPACE

• 成果类型: Article
• 署名作者: Mussel, Elchanan; Neeman, Joe
• 署名单位: University of California System; University of California Berkeley
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/13-AOP860
• 发表日期: 2015
• 页码: 971-991
• 关键词: semigroup proofs; noise stability; INEQUALITY; optimality

摘要

We prove the first robust dimension free isoperimetric result for the standard Gaussian measure gamma n and the corresponding boundary measure gamma n(+) in R-n. The main result in the theory of Gaussian isoperimetry (proven in the 1970s by Sudakov and Tsirelson, and independently by Borell) states that if gamma n (A) = 1/2 then the surface area of A is bounded by the surface area of a half-space with the same measure, gamma n(+) (A) <= (2 pi)(-1/2). Our results imply in particular that if A subset of R-n satisfies gamma n (A) = 1/2 and gamma n(+) (A) <= (2 pi)(-1/2) + delta then there exists a half-space B subset of R-n such that gamma n (A,A,B) <= Clog(-1/2)(1/delta) for an absolute constant C. Since the Gaussian isoperimetric result was established, only recently a robust version of the Gaussian isoperimetric result was obtained by Cianchi et al., who showed that gamma n (A Delta B) <= C(n)root delta for some function C(n) with no effective bounds. Compared to the results of Cianchi et al., our results have optimal (i.e., no) dependence on the dimension, but worse dependence on delta.