# Gaussian concentration to upper bound the variance of the norm of a gaussian vector

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?