Let $f : \mathbb{R}^n \to \mathbb{R}$ be a convex function and let $c$ be some constant. Show that the following set $$s= \{x \in \mathbb{R}^n \mid f(x) \le c \}$$ is convex.
Looking for a hint.
Let $f : \mathbb{R}^n \to \mathbb{R}$ be a convex function and let $c$ be some constant. Show that the following set $$s= \{x \in \mathbb{R}^n \mid f(x) \le c \}$$ is convex.
Looking for a hint.
Hint: Well, just write down a convex combination of elements in $s$ and verify that it belong to $s$. You will find the convexity of $f$ useful for this.