Let $X \subseteq \mathbb R^n$ be a convex set and $g\colon X\to\mathbb R$ a concave function. Prove that the set $\{(x,z)\in X \times \mathbb R \mid g(x) \geq z\}$ is a convex set.
How do I go about proving this?
If I take 2 elements in $X \times \mathbb R$, e.g. $(x_1,z)$ and $(x_2,z)$ I will end up with an $g(z)$ and can't prove the inequality.
Thanks.