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 \(\gamma _n\). For any set \(A\subseteq \mathbb R^n\) and real \(t\), its rotation sensitivity is the real value of the product-measure mass of the event \(\mathbf1_A(G)\ne \mathbf1_A(\cos t\, G+\sin t\, G\prime )\); for measurable \(A\), this is the disagreement probability.

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 \(A\subseteq \mathbb R^n\), its Gaussian surface area is the extended nonnegative quantity \(P_\gamma (A)=\liminf _{t\to 0^+}[\sqrt{\pi /2}\, s_A(t)/t]_+\), where \([u]_+=\max (u,0)\).

For a real number \(\alpha \), the closed Gaussian profile is \(U(\alpha )=\varphi (\Phi ^{-1}(\alpha ))\) if \(0\lt \alpha \lt 1\), and \(U(\alpha )=0\) otherwise, where \(\varphi \) and \(\Phi \) are the standard Gaussian density and distribution function.