This section is devoted to studying scifinder the Gaussian Isoperimetric Inequality . This inequalit

Contents scifinder Book (91) All chapter sections (56) Chapter 01: Boolean functions and the Fourier expansion (8) Chapter 02: Basic concepts and social choice (9) Chapter 03: Spectral structure and learning (8) Chapter 04: DNF formulas and small-depth circuits (8) Chapter 05: Majority and threshold functions (9) Chapter scifinder 06: Pseudorandomness and ${\mathbb F}_2$-polynomials (8) Chapter 07: Property testing, PCPPs, and CSPs (7) Chapter 08: Generalized domains scifinder (10) Chapter 09: Basics of hypercontractivity (9) Chapter 10: Advanced hypercontractivity (8) Chapter 11: Gaussian space and Invariance Principles (5) Chapter notes (10) Chapter scifinder overview (11) Deleted scifinder scenes (1) Exercises (13) CMU Online Course (23) Simons Symposium (16) Uncategorized (6)
This section is devoted to studying scifinder the Gaussian Isoperimetric Inequality . This inequality is a special case of the Borell Isoperimetric Inequality scifinder (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 we have that for any $A \subseteq {\mathbb R}^n$, if $H \subseteq {\mathbb R}^n$ is a halfspace with the same Gaussian volume as $A$ then for all $\epsilon$, \[ \mathbf{RS}_A(\epsilon) \geq \mathbf{RS}_H(\epsilon). \] Since $\mathbf{RS}_A(0) = \mathbf{RS}_H(0) = 0$, it follows that \[ \mathbf{RS}_A'(0^+) \geq \mathbf{RS}_H'(0^+). \] (Here we are considering the one-sided derivatives at $0$, which can be shown to exist, though $\mathbf{RS}_A’(0^+)$ may equal $+\infty$; see the notes at the end of this chapter.) As will be explained shortly, $\mathbf{RS}_A’(0^+)$ is precisely $\sqrt{2/\pi} \cdot \text{surf}_\gamma(A)$, where $\text{surf}_\gamma(A)$ denotes the “Gaussian surface area” of $A$. Therefore the above inequality is equivalent to the following:
Gaussian Isoperimetric Inequality Let $A \subseteq {\mathbb R}^n$ have $\mathrm{vol}_\gamma(A) = \alpha$ and let $H \subseteq {\mathbb R}^n$ be any halfspace with $\mathrm{vol}_\gamma(H) = \alpha$. Then $\text{surf}_\gamma(A) \geq \text{surf}_\gamma(H)$.
Remark 47 As shown in Proposition 49 below, the right-hand side in this inequality is equal to $\mathcal{U}(\alpha)$, where $\mathcal{U}$ is the Gaussian isoperimetric function , encountered scifinder earlier in Definition 5.23 and defined by $\mathcal{U} = \varphi \circ \Phi^{-1}$.
Let’s now discuss the somewhat technical question of how to properly define $\text{surf}_\gamma(A)$, the Gaussian surface scifinder area of a set $A$. Perhaps the most natural definition would be to equate it with the Gaussian Minkowski content of the boundary $\partial A$ of $A$, \begin{equation} \label{eqn:heuristic-gsa} \gamma^+(\partial A) = \liminf_{\epsilon \to 0^+} \frac{\mathrm{vol}_\gamma(\{z : \mathrm{dist}(z, \partial A) < \epsilon/2\})}{\epsilon}. \end{equation} (Relatedly, one might also consider the surface integral over $\partial A$ of the Gaussian pdf $\varphi$.) Under the “official” definition of $\text{surf}_\gamma(A)$ we give below in Definition 48 , we’ll indeed have $\text{surf}_\gamma(A) = \gamma^+(\partial A)$ whenever $A$ is sufficiently nice — say, a disjoint union of closed, scifinder full-dimensional, scifinder convex sets. However, the Minkowski content definition is not a good one in general because it’s possible to have $\gamma^+(\partial A_1) \neq \gamma^+(\partial A_2)$ for some sets $A_1$ and $A_2$ that are equivalent up to measure $0$. (For more information, see an exercise and the notes at the end of this chapter.) As mentioned above, one “correct” scifinder definition is $\text{surf}_\gamma(A) = \sqrt{\pi/2} \cdot \mathbf{RS}_A’(0^+)$. This definition has the advantage of being insensitive to measure-$0$ changes to $A$. To connect this unusual-looking definition with Minkowski content, let’s heuristically interpret $\mathbf{RS}_A’(0^+)$. We start by thinking of it as $\frac{\mathbf{RS}_A(\epsilon)}{\epsilon}$ for “infinitesimal $\epsilon$”. Now $\mathbf{RS}_A(\epsilon)$ can be thought of as the probability that the line segment $\boldsymbol{\ell}$ joining two $\cos \epsilon$-correlated Gaussians crosses $\partial A$. Since $\sin \epsilon \approx \epsilon$, $\cos \epsilon \approx 1$ up to $O(\epsilon^2)$, we can think of these correlated Gaussians as $\boldsymbol{g}$ and $\boldsymbol{g} + \epsilon \boldsymbol{g}’$ for independent $\boldsymbol{g}, \boldsymbol{g}’ \sim \mathrm{N}(0,1)^n$. When $\boldsymbol{g}$ lands near $\partial A$, the length of $\boldsymbol{\ell}$ in the direction perpendicular to $\partial scifinder A$ will, in expectation, be $\epsilon \mathop{\bf E}[|\mathrm{N}(0,1)|] = \sqrt{2/\pi} \epsilon$. Thus $\mathbf{RS}_A(\epsilon)$ should essentially be $\sqrt{2/\pi} \epsilon \cdot \mathrm{vol}_\gamma(\{z : \mathrm{dist}(z, \partial A) < \epsilon/2\})$ and we have heuristically justified \begin{equ