superlevel sets of a concave function are convex

Question

I'm reading a paper where it's written that the superlevel sets of a concave function are convex. First what's a "super"level? And why is this true ? Thank you.

Answer 1

Let $f : X \to \mathbb{R}$. The superlevel set $S(\alpha) \subset X$ is the set of values in the domain for which the function value is at least $\alpha$: $$S(\alpha) = \{ x \in X : f(x) \geq \alpha\}. $$ This set is indeed convex, which can be proven using the concavity of $f$. Let $\alpha \in \mathbb{R}$, $x_1,x_2 \in S(\alpha)$, and $\lambda \in [0,1]$, then: $$f(\lambda x_1 + (1-\lambda) x_2) \geq \lambda f(x_1) + (1-\lambda) f(x_2) \geq \lambda \alpha + (1-\lambda)\alpha = \alpha.$$ Therefore, $\lambda x_1 + (1-\lambda) x_2 \in S(\alpha)$.