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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.

Rodrigo de Azevedo
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GBa
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1 Answers1

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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.

Rodrigo de Azevedo
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Rasmus
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    Note: I didn't answer the question completely since it could be homework and since the OP asked for a hint. I hope that it is appropriate to post this as an answer. – Rasmus Sep 29 '10 at 17:15
  • So every point in between points of the set can be written as: a1x1 + a2x2 + ...anxn where every ai >= 0 and the sum of the a values is 1. How can it be shown this belongs to s? – GBa Sep 29 '10 at 21:10
  • @Greg: What does it mean to belong to s? – Rasmus Sep 30 '10 at 08:49
  • if an element is in s then f (λx1 + (1 − λ)x2 ) ≤ λf (x1 ) + (1 − λ)f (x2 ) which is ≤ c .... I'm just confused how c fits in with this problem. – GBa Sep 30 '10 at 16:56
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    @Greg: there you have the proof: if x1 and x2 are in s then the line segment they define is totally contained in s, and so s is convex, by definition. – lhf Sep 30 '10 at 18:17