Gaussian integral representation of the norm

For $v$ in a finite-dimensional real inner product space $E$, $\int_E |\langle v, w\rangle|\,d\gamma(w) = \|v\| \int_{\mathbb{R}} |s|\,dN(0,1)(s)$, where $\gamma$ is the standard Gaussian measure on $E$. The key analytic input for the $p=2$ Hlawka proof: it reduces the norm inequality to the one-dimensional case, bypassing the box-SOS method which is provably powerless at $p=2$.