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Consider that we have a set of non-convex functions $f_1, f_2, ...., f_n$. Is it also non-convex the multiplication of these functions? In other word, $f=f_1 \times f_2 \times...\times f_n$ is non-convex? Have anyone a reference article or book for this?

mohammad
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For a counterexample, let $f_1(x)=f_2(x)=\sqrt x$.

See this answer: Proving that multiplication of convex function is convex

  • Consider that the domain of $x$ in $f_1$ and $f_2$ is not negative. Therefore, the domain of $x$ is not negative and we can conclude that your example is not convex. The link is the inverse of my question. – mohammad Nov 22 '19 at 11:35
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    Here is another example: $f_1(x)=f_2(x)=x^{2/3}$. The product is $f(x)=x^{4/3}$ which is convex.

    Also, I know my link is the inverse but maybe you can get some ideas there.

     –  Nov 23 '19 at 18:34
  • Another source of information on this topic: https://www.ima.umn.edu/sites/default/files/2204.pdf – stewori Apr 18 '20 at 21:18