Currently I'm reading a book on asymptotic geometric theory, and in the book, the author mentions the following situation: Let $K \subset \mathbb{R}^n$ be a origin symmetric convex body and consider the space $X = (\mathbb{R}^n, || \cdot||)$, where the norm $||\cdot||$ is such that $K$ is the unit ball of $X$ and, for each $q \geq 1$ define $$ M_q = M(X) = \left( \int_{S^{n-1}} ||x||^q d \sigma(x) \right)^{1/q}, $$ where $S^{n-1}$ is the Euclidean unit sphere, and $\sigma$ is the Haar measure on $S^{n-1}$. What follows is my question:
Let $g_1, \dots g_n$ be independent standard Gaussian random variables on a probability space $\Omega$ and $e_1',\dots,e_n'$ be an orthonormal basis for $\mathbb{R}^n$. It is stated that integration in polar coordinates who that $$ \left( \int_{\Omega} \left| \left| \sum_{i=1}^n g_i(\omega) e_i' \right| \right|^2 d \omega \right)^{1/2} = \sqrt{n} M_2. $$
I'm unsure of how to get this identity. I tried working in two variables and using independence, but I couldn't get it even in this case. Does anyone see where this is coming from?
The vector $z=\sum_{i=1}^ng_i(\omega)e_i'$ has an $n$-dimensional standard gaussian distribution, so can be represented as $z=x t$ where $x$ is uniformly distributed over the unit sphere $S^{n-1}$ and $t^2$ has the $\chi^2_n$ distribution and where $x$ and $t$ are independent. The quantity $Q=\int_\Omega \|z\|_K^2\,d\omega$ can thus be written as $$Q= \int_{S^{n-1}} \int_0^\infty t^2 \|x\|_K^2\, dt\, d\sigma(x) = (Et^2)\int_{S^{n-1}} \|x\|_K^2d\sigma(x)=n\, M_2.$$