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I'd like to know how one would go about showing that the following function, $f$, that is almost everywhere positive exists:

$$g(x_1,\cdots,x_n)=\int_{-\infty}^{\infty} \prod _{j=1}^nf(u-x_j)du$$ where $g:\mathbb{R}^n:\rightarrow \mathbb{R}$ satisfies:

(1) $g(x_1,\cdots,x_n)$ is the derivative is the nth order partial derivative of $\frac{\partial \ln(G(e^{x_1},\cdots,e^{x_n})}{\partial x_1x_2...x_n}$; where $G(e^{x_1},\cdots,e^{x_n})$ is symmetric, homogenous of degree 1, $G(0)=0$, $\lim G(y)\rightarrow \infty$ as $y \rightarrow \infty$, $G(y)>0$.

(2) $g\ge0$

If this isn't possible, can you suggest of ways to add restrictions so that such an $f$ exists? I know this is asking a lot, but I was wondering if someone would be willing to give some direction.

Thanks so much in advance!!!!!

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    Did you try in $\mathbb R^2$ with $g(x,y)=(x^2+y^2)^{-1/2}$?2012-03-01
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    @DavideGiraudo I think he wants to show the existence of $f$ given $g$.2012-03-01
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    I think it's indeed what the OP wants, but I don't think there is $f$ in this case.2012-03-01
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    You guys are right. I'm relaxing the homogeneity assumption.2012-03-01

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