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Here's a homework question I'm struggling with:

Prove/disprove the next statement:

Let $f,g$ two convex functions, then $h(x)=f(x) \cdot g(x)$ is also convex

So, we know that $h'(x)=f'(x) \cdot g(x) + f(x) \cdot g'(x)$. We also know that $f'(x),g'(x)$ are monotonically increasing because they are convex. If I can show that $h'(x)$ is also monotonically increasin I'm done, but I'm not sure how to do it. Any hints?


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Try to find a counterexample ... :D – martini May 21 '12 at 8:14
That didn't help me too mcuh, sorry – yotamoo May 21 '12 at 8:25
Ok ... $x \mapsto (x-a)^2$ is convex for any $a$, right :) – martini May 21 '12 at 8:28
up vote 6 down vote accepted

Hint: You can write the function defined by $x\mapsto-x^2$ as the product of two very simple linear and hence convex functions.

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So the statement is not even true. I assumed it is. Thanks! – yotamoo May 21 '12 at 8:37
I am sorry but I am new to convex functions. Is the statement "the product of convex functions is convex" true or false ? Could you please clarify and give more details. For example if $f(x)=-x^2$ and $g(x)=-x^2$ then $fg$ is a convex function whereas $f$ and $g$ are not. – roni Aug 25 '14 at 7:41
@roni It is false. Take the two functions $x\mapsto x$ and $x\mapsto -x$. – Michael Greinecker Aug 25 '14 at 16:40

The functions $f(x)=1-x$ and $g(x)=1+x$ are convex. However, their product $(f*g)(x)=1-x^2$ is not. So it hoes not hold.

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