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I have a smooth function $g(x) \colon \mathbb{R}^n \to \mathbb{R}$ that is homogeneous of order one: $g(\lambda x) = \lambda g(x)$. Let $(x_1,...,x_n) \circ (y_1,...,y_n) = (x_1 y_1,...,x_n y_n)$. Are there some sufficient conditions for $$ h(x,y) = \det \left\| \frac{\partial^2 g(x \circ y)}{\partial x_i \partial y_j} \right\| > 0 $$ I found that $$ h(x,y) = \det \left( \left\| \partial_{i,j} g(x \circ y) x_i y_j \right\| + \left\| \partial_{i} g(x \circ y) \delta_{i,j} \right\| \right) $$ where $\partial_i g(x) = \frac{\partial g(x)}{\partial x_i}$.

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Did you try the case $n=1$? –  Ali Feb 12 '12 at 13:27
This case isn't interesting, because in this case $g(x) = xg(1)$ and $g(x)$ is linear. –  Nimza Feb 12 '12 at 13:36
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