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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.

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    Take $0\leq \alpha\leq 1$, put $x':=\alpha x_1+(1-\alpha)x_2$ and show that $g(x')\geq z$, using the concavity of $g$.2011-10-06

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