Problem in convex analysis

Question

I found this problem in one of the old exams for convex analysis:

Let $A \subseteq \mathbb{R}^n$ be a convex set and $f:A \rightarrow \mathbb{R}$ a convex function.
a) Show that $f^{-1}(-\infty,a)$ is a convex set for every $a \in \mathbb{R}$.
b) Find an example when $f^{-1}(0,\infty)$ is not a convex set.

Could someone help me with this problem please?

score 0 · Accepted Answer · answered Nov 08 '13 at 14:23

0

Some hints:

for $a)$: $f^{-1}(-\infty,a)=\{x\in A|f(x)< a\}=:C_a$. To show convexity, we have to verify, for $x,y\in C_a$ $\lambda x+(1-\lambda) y\in C_a$, where $\lambda\in [0,1]$. Use the convexity of $f$.

for $b)$: Have a look at $A=(-4,4)$, $f=x^2-4$. In this example $f^{-1}(0,\infty)=(-4,-2)\cup (2,4)$ and this is clearly not a convex set

answered Nov 08 '13 at 14:23

math

4,347

Thank you for the hints. They helped me to get to the end of the proof. – eta Nov 09 '13 at 12:01
@eta you're welcome – math Nov 09 '13 at 12:33

Problem in convex analysis

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