## §11.4: Gaussian surface area and Bobkov’s Inequality

This section is devoted to studying the Gaussian Isoperimetric Inequality. This inequality is a special case of the Borell Isoperimetric Inequality (and hence also a special case of the General-Volume Majority Is Stablest Theorem); in particular, it’s the special case arising from the limit $\rho \to 1^{-}$.

Restating Borell’s theorem using rotation sensitivity [...]

## §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.

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## §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 $f$ [...]

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## §10.5: Highlight: General sharp threshold theorems

In Chapter 8.4 we described the problem of “threshold phenomena” for monotone functions $f : \{-1,1\}^n \to \{-1,1\}$.

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## §10.4: More on randomization/symmetrization

In Section 3 we collected a number of consequences of the General Hypercontractivity Theorem for functions $f \in L^2(\Omega^n, \pi^{\otimes n})$. All of these had a dependence on “$\lambda$”, the least probability of an outcome under $\pi$. This can sometimes be quite expensive; for example, the KKL Theorem and its consequence Theorem 28 are trivialized [...]

## §10.3: Applications of general hypercontractivity

In this section we will collect some applications of the General Hypercontractivity Theorem, including generalizations of the facts from Section 9.5.

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