I want to prove the following:
Let $X_1$ be a centered Gaussian vector in $\mathbb{R}^d$ with covariance matrix $\Sigma$. Then: $ Var( \|X_1 \|) \lesssim \left(\mathbb{E} \|X_1 \| \right)^2 $
This is taken from https://arxiv.org/pdf/1610.05200.pdf (proof of Corollary 5.7).
The idea is to use Gaussian concentration (Lemma 3.8 in the same file). Using, then Lemma 3.8 I obtain: $$ \mathbb{E} \left[\left( \|X_1\| - \mathbb{E} \|X_1\| \right)^2 \right] = Var(\|X_1\|) \lesssim 2 \mathbb{E} \| \nabla\left( \|X_1\| \right)\|^2 $$ Then, I just need to bound: $\mathbb{E} \| \nabla\left( \|X_1\| \right)\|^2$. In order to do so, from what they say in the paper, the strategy to be adopted is to study the function $f(X_1) = \| X_1 \|$. This functions should be 1-Lipschitz, then the its gradient is bounded (Lipschitz implies bounded gradient) This would allow me to say that $ Var(\|X_1\|) \lesssim 1 $ but to conclude and prove the first statement I would need to additionally prove something like $\mathbb{E} \| X_1 \| \gtrsim 1 $ which I'm failing to see.
I believe that I'm probably making some mistakes when analyzing the Lipschitzness property of the function $\| \cdot \| $. Do you have any idea on how to prove this statement needed for Corollary 5.7 in the pdf?
