Let $f\colon \mathbb{R}^n \to \mathbb{R}$ be a differentiable function. I am looking for conditions under which the function $$x\mapsto \Vert\nabla f(x)\Vert^2$$ is convex.
It obviously hold if $f$ is quadratic, and therefore $\nabla f$ is linear. So I am wondering whether it holds for functions with Lipschitz gradients?
Every norm $||\cdot||$ is convex. The quadratic function $x^2$ is non-decreasing in $x\ge0$, so the squared norm $||\cdot||^2$, i.e. the composition, is convex. There are no specific conditions that you need to impose.