## §11.3: Borell’s Isoperimetric Theorem

If we believe that the Majority Is Stablest Theorem should be true, then we also have to believe in its “Gaussian special case”. Let’s see what this Gaussian special case is. Suppose $f : {\mathbb R}^n \to [-1,1]$ is a “nice” function (smooth, say, with all derivatives bounded) having $\mathop{\bf E}[f] = 0$. You’re [...]

## §11.2: Hermite polynomials

Having defined the basic operators of importance for functions on Gaussian space, it’s useful to also develop the analogue of the Fourier expansion.

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## §11.1: Gaussian space and the Gaussian noise operator

We begin with a few definitions concerning Gaussian space.

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## Chapter 11: Gaussian space and Invariance Principles

The final destination of this chapter is a proof of the following theorem due to Mossel, O’Donnell, and Oleszkiewicz [MOO05a,MOO10], first mentioned in Chapter 5.25:

Majority Is Stablest Theorem Fix $\rho \in (0,1)$. Let $f : \{-1,1\}^n \to [-1,1]$ have $\mathop{\bf E}[f] = 0$. Then, assuming $\mathbf{MaxInf}[f] \leq \epsilon$, or more generally that $f$ [...]