= Gaussian rotation interpolation inequality
{c}
{title2=$\mathbb E\Psi(f(X)-\mathbb Ef(X))\leq\mathbb E\Psi(\tfrac\pi2\langle\nabla f(X),Y\rangle)$}
For <independent and identically distributed random variables> $X,Y$ with a centered <multivariate normal distribution>, a <continuously differentiable function> $f$ with bounded <gradient>, and a <convex function> $\Psi$ for which the expressions are integrable, the displayed inequality holds. First apply <Jensen inequality> conditionally to $f(X)-f(Y)$. For the <Gaussian rotation of independent copies>, the <chain rule> gives $f(X)-f(Y)=\int_0^{\pi/2}\langle\nabla f(U_\theta),V_\theta\rangle\,d\theta$. A second application of <Jensen inequality> to the uniform measure on $[0,\pi/2]$, followed by the <Gaussian rotation of independent copies>, proves the inequality. The useful choice $\Psi(v)=e^{\lambda|v|}$ yields a <Gaussian concentration inequality> through an <exponential moment of an absolute standard normal variable>.
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