I am currently trying to prove the convexity of the following function of x, and I unfortunately have problems finding a solution
$$ \sqrt{\mathbf{x^{\top}}\mathbf{Q} \mathbf{x} + 2(\boldsymbol{\sigma_1} - \sigma_{nn} \mathbf{1_1})^{\top}\mathbf{x} + \sigma_{nn}}, $$
with $x, \sigma_1 \in \mathbb{R}^{n-1}$ and $\sigma_{nn} \in \mathbb{R}$ and $1_1 \in \mathbb{R}^{n-1}$ the vector consisting only of ones, $\mathbf{Q}$ a positive definite matrix. Does somebody maybe have an idea how to solve this? I would be very thankful!