triyin to solve this exercise, any advices and hints on how to prove it are more than helpful.
The problem
Let $C \subset \mathbb{R}^n$ be a convex set, and $f: C \rightarrow \mathbb{R}$ is also convex. Show that the set $\Gamma \subset C$, where $f$ reach it's minimal value, is convex.
Take two points A and B where the min is achieved and let P be in the segment AB. By convexity $f(P)$ is below a convex combination of $f(A)$ and $f(B)$ which is just $f(A)$. On the other hand you know $f(P)\ge f(A)$ and you are done.