Show that the set $\{x\in\mathbb{R}^N:\nabla f(x)=0 \}$ is convex

Question

Let $f:\mathbb{R}^N\rightarrow \mathbb{R}$ be a $C^1$ convex function. Show that $\{x\in\mathbb{R}^N:\nabla f(x)=0 \}$ is convex (we assume that empty set is convex).

Any hint?

Answer 1

Convexity implies $f(y)\ge f(x)+\langle \nabla f(x),y-x\rangle$ for all $x,y$. Specializing this to the points of your set, you will find they are points where $f$ attains its global minimum, say $m$. Argue that $\{x:f(x)=m\}$ is convex.