Robust Dimension Free Isoperimetry in Gaussian Space

Details Robust Dimension Free Isoperimetry in Gaussian Space Robust Dimension Free Isoperimetry in Gaussian Space

2012-02-19

arxiv.org/abs/1202.4124v2

Mathematics

Probability

Scientific paper

We prove a robust dimension free isoperimetric result for the standard Gaussian measure \gamma_n and the corresponding boundary measure \gamma_n^+ in R^n. Our results imply in particular that if A \subset R^n satisfies \gamma_n(A) = 1/2 and \gamma_n^+(A) \leq \gamma_n^+(B)+\delta, where B = \{x : x_1 \leq 0\} is a half-space, then there exists a half-space (a rotation of B) B' such that \gamma_n(A \Delta B') \leq c \log^{-1/6} (1/\delta) for an absolute constant c. Compared to recent results of Cianchi et al., who showed that \gamma_n(A \symdiff B') \le c(n) \sqrt \delta where $c(n)$ grows polynomially with n, our results have better (i.e. no) dependence on the dimension, but worse dependence on \delta.

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