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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.

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    So$r$ry for the editing confusion. I realise I don't know how latex works here.2010-09-30

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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.

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    @Gr$e$g: th$e$re you have the proof: if x1 a$n$d x2 are i$n$ s then the line segment they define is totally contained in s, and so s is convex, by definition.2010-09-30