Suppose I have a smooth function $f:\mathbb{R}^d\mapsto \mathbb{R}$. The image of the gradient filed of $f$ is defined as $$V = \{v\in \mathbb{R}^d:\exists x, s.t. \nabla f(x) = v\}.$$ Are there some general theory or technical tools that we can use to prove that $V$ is a convex set? I have searched a lot but didn't find any. Any comments would be appreciated!
I asked a related question here, which is equivalent to the case for a special $f$.