I have a convex function $f:R_{\geq0}^N\rightarrow R$ and its first order derivatives are all positive. Now I find it is difficult to prove that the convexity of the function $g(x,y)=xf(y)$, where $x\in R_{\geq0}$ and $y\in R^N_{\geq0}$. Anybody knows how to do it?
Is the product of a positive variable and a convex function also convex?
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convex-analysis
convex-optimization
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1I'm curious what led you to believe this might be true. It is not. – 2017-02-01
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0I think you also need $f(x) \geq 0$ for your statement to hold. – 2017-02-01
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0It does not matter whether or not $f(x)\geq 0$, it does not hold. – 2017-02-01
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3Just look at the Hessian of $g(x,y)=xy$. – 2017-02-01