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I want to check if the following function is convex with respect to the vector variable $x$.

$$ R(x) = \log_2 \left( \sum_{i=1}^M {\frac{p}{((x_i-\gamma)^2 + (y_i-\beta)^2 +(z_i-\rho)^2)^\alpha}+\sigma^2} \right) $$

I have tried to check it by myself using graphing and it seems to me that it is convex, but I can not be sure if that is true from the point of view of convexity analysis.

Is there a good way to check that with analysis or a good reference to help me?

Please assume that all other variables be fixed.

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  • $\begingroup$ Hello and welcome to math.stackexchange. Please be a little more precise: What is $\alpha$? Are the $y_i$ and $z_i$ fixed parameters? As written, the function is not convex even in the case $M = 1$. Take e.g. $\gamma = 0, \alpha = \sigma = 1$ and the other parameters arbitrary such that $y_1 \ne \beta$ and see for yourself. $\endgroup$ Jan 7 at 16:16
  • $\begingroup$ @HansEngler yes, alpha and all other parameters are assumed fixed. $\endgroup$ Jan 7 at 16:59

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Let $\mathbf{x}_0 = (\gamma, \gamma, \dots, \gamma)$. We have $R(\mathbf{x}_0) = \log_2 \left(\sigma^2 + A \right)$ for some positive $A$ and also $$ \lim_{\mathbf{x} \to \infty} R(\mathbf{x}) = \log_2 \sigma^2 < R(\mathbf{x}_0)\, . $$ In fact $R$ has a global maximum at $\mathbf{x}_0$. Such a function cannot be convex.

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