252 Boolean Analysis — SurfaceArea
252.1 Overview
The rotation definition gives an extended Gaussian perimeter, with a genuine right limit, null-set invariance, halfspace values, and isoperimetric bounds.
252.2 Declarations
Let \(f\colon \mathbb R\to \mathbb R\) be continuous at zero with \(f(0)=0\) and \(f(s+t)\le f(s)+f(t)\) for all real \(s,t\). Then for every \(\delta \gt 0\), \([f(\delta )/\delta ]_+\le \liminf _{t\to 0^+}[f(t)/t]_+\) in \([0,\infty ]\), where \([u]_+=\max (u,0)\).
Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). Write \(s_A(t)\) for the real value of the product-measure mass \((\gamma _n\otimes \gamma _n)\{ (g,h):\mathbf1_A(g)\ne \mathbf1_A(\cos t\, g+\sin t\, h)\} \). Write \(P_\gamma (A)=\liminf _{t\to 0^+}\bigl[\sqrt{\pi /2}\, s_A(t)/t\bigr]_+\) in \([0,\infty ]\), where \([x]_+=\max (x,0)\). If \(A\subseteq \mathbb R^n\) is measurable and \(\delta \gt 0\), then \([\sqrt{\pi /2}\, s_A(\delta )/\delta ]_+\le P_\gamma (A)\).
Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). Write \(s_A(t)\) for the real value of the product-measure mass \((\gamma _n\otimes \gamma _n)\{ (g,h):\mathbf1_A(g)\ne \mathbf1_A(\cos t\, g+\sin t\, h)\} \). Write \(P_\gamma (A)=\liminf _{t\to 0^+}\bigl[\sqrt{\pi /2}\, s_A(t)/t\bigr]_+\) in \([0,\infty ]\), where \([x]_+=\max (x,0)\). For every measurable \(A\subseteq \mathbb R^n\), \([\sqrt{\pi /2}\, s_A(t)/t]_+\to P_\gamma (A)\) as \(t\to 0^+\), with convergence in \([0,\infty ]\).
Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). Write \(s_A(t)\) for the real value of the product-measure mass \((\gamma _n\otimes \gamma _n)\{ (g,h):\mathbf1_A(g)\ne \mathbf1_A(\cos t\, g+\sin t\, h)\} \). For any sets \(A,B\subseteq \mathbb R^n\) with \(\mathbf1_A=\mathbf1_B\) almost everywhere under \(\gamma _n\), \(s_A(t)=s_B(t)\) for every real \(t\).
Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). Write \(s_A(t)\) for the real value of the product-measure mass \((\gamma _n\otimes \gamma _n)\{ (g,h):\mathbf1_A(g)\ne \mathbf1_A(\cos t\, g+\sin t\, h)\} \). Write \(P_\gamma (A)=\liminf _{t\to 0^+}\bigl[\sqrt{\pi /2}\, s_A(t)/t\bigr]_+\) in \([0,\infty ]\), where \([x]_+=\max (x,0)\). If measurable sets \(A,B\subseteq \mathbb R^n\) agree almost everywhere under \(\gamma _n\), then \(P_\gamma (A)=P_\gamma (B)\).
Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). Write \(s_A(t)\) for the real value of the product-measure mass \((\gamma _n\otimes \gamma _n)\{ (g,h):\mathbf1_A(g)\ne \mathbf1_A(\cos t\, g+\sin t\, h)\} \). Write \(P_\gamma (A)=\liminf _{t\to 0^+}\bigl[\sqrt{\pi /2}\, s_A(t)/t\bigr]_+\) in \([0,\infty ]\), where \([x]_+=\max (x,0)\). For a vector \(v\in \mathbb R^n\) with \(\sum _i v_i^2=1\) and real \(t\), the halfspace \(H=\{ z:\sum _i v_i z_i\le t\} \) satisfies \(P_\gamma (H)=\varphi (t)\), where \(\varphi (t)=(2\pi )^{-1/2}e^{-t^2/2}\).
Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). Write \(s_A(t)\) for the real value of the product-measure mass \((\gamma _n\otimes \gamma _n)\{ (g,h):\mathbf1_A(g)\ne \mathbf1_A(\cos t\, g+\sin t\, h)\} \). Write \(P_\gamma (A)=\liminf _{t\to 0^+}\bigl[\sqrt{\pi /2}\, s_A(t)/t\bigr]_+\) in \([0,\infty ]\), where \([x]_+=\max (x,0)\). For every measurable \(A\subseteq \mathbb R^n\) and \(\delta \gt 0\), \([\sqrt{\pi /2}\, s_A(\delta )/\delta ]_+\le P_\gamma (A)\).
Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). Write \(s_A(t)\) for the real value of the product-measure mass \((\gamma _n\otimes \gamma _n)\{ (g,h):\mathbf1_A(g)\ne \mathbf1_A(\cos t\, g+\sin t\, h)\} \). Write \(P_\gamma (A)=\liminf _{t\to 0^+}\bigl[\sqrt{\pi /2}\, s_A(t)/t\bigr]_+\) in \([0,\infty ]\), where \([x]_+=\max (x,0)\). If \(A\subseteq \mathbb R^n\) is measurable with \(\alpha =\gamma _n(A)\), then \(2\sqrt{2/\pi }\, \alpha (1-\alpha )\le P_\gamma (A)\).
Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). Write \(s_A(t)\) for the real value of the product-measure mass \((\gamma _n\otimes \gamma _n)\{ (g,h):\mathbf1_A(g)\ne \mathbf1_A(\cos t\, g+\sin t\, h)\} \). Write \(P_\gamma (A)=\liminf _{t\to 0^+}\bigl[\sqrt{\pi /2}\, s_A(t)/t\bigr]_+\) in \([0,\infty ]\), where \([x]_+=\max (x,0)\). Let \(A\subseteq \mathbb R^n\) be measurable and \(f\colon \mathbb R^n\to [0,1]\) be differentiable with continuous derivative \(D(x)\). If \(\mathbb E|f(G)-f(\cos \theta \, G+\sin \theta \, G\prime )|\le s_A(\theta )\) for every real \(\theta \), then \(\int \| \nabla f(x)\| _2\, d\gamma _n(x)\le P_\gamma (A)\) as a nonnegative extended integral.
Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). Write \(s_A(t)\) for the real value of the product-measure mass \((\gamma _n\otimes \gamma _n)\{ (g,h):\mathbf1_A(g)\ne \mathbf1_A(\cos t\, g+\sin t\, h)\} \). Write \(P_\gamma (A)=\liminf _{t\to 0^+}\bigl[\sqrt{\pi /2}\, s_A(t)/t\bigr]_+\) in \([0,\infty ]\), where \([x]_+=\max (x,0)\). Let \(A\subseteq \mathbb R^n\) be measurable, \(0\lt \alpha \lt 1\), and let \(U\) be the standard Gaussian isoperimetric profile extended by zero outside \((0,1)\). Set \(f_u(x)=\mathbb E[\mathbf1_A(\cos u\, x+\sin u\, G)]\). If \(U(\alpha )\le [\int U(f_u)\, d\gamma _n]_++P_\gamma (A)\) for every \(0\lt u\lt \pi /2\), then \(U(\alpha )\le P_\gamma (A)\).
Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). Write \(s_A(t)\) for the real value of the product-measure mass \((\gamma _n\otimes \gamma _n)\{ (g,h):\mathbf1_A(g)\ne \mathbf1_A(\cos t\, g+\sin t\, h)\} \). Write \(P_\gamma (A)=\liminf _{t\to 0^+}\bigl[\sqrt{\pi /2}\, s_A(t)/t\bigr]_+\) in \([0,\infty ]\), where \([x]_+=\max (x,0)\). Let \(A\subseteq \mathbb R^n\) be measurable with \(\gamma _n(A)=\alpha \in (0,1)\) and put \(f_u(x)=\mathbb E[\mathbf1_A(\cos u\, x+\sin u\, G)]\). Suppose that for each \(0\lt u\lt \pi /2\), \(f_u\) has a derivative family \(D_u\) and there exist constants \(C_u\ge 0\) and \(M_u\in \mathbb R\) such that \(D_u\) is \(C_u\)-Lipschitz and \(\| D_u(x)\| \le M_u\) for all \(x\). Then \(\varphi (\Phi ^{-1}(\alpha ))\le P_\gamma (A)\), where \(\varphi \) and \(\Phi \) are the standard Gaussian density and distribution function.
Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). Write \(s_A(t)\) for the real value of the product-measure mass \((\gamma _n\otimes \gamma _n)\{ (g,h):\mathbf1_A(g)\ne \mathbf1_A(\cos t\, g+\sin t\, h)\} \). Write \(P_\gamma (A)=\liminf _{t\to 0^+}\bigl[\sqrt{\pi /2}\, s_A(t)/t\bigr]_+\) in \([0,\infty ]\), where \([x]_+=\max (x,0)\). For every measurable \(A\subseteq \mathbb R^n\) of Gaussian volume \(\alpha \in (0,1)\), \(\varphi (\Phi ^{-1}(\alpha ))\le P_\gamma (A)\), where \(\varphi \) and \(\Phi \) are the standard Gaussian density and distribution function.