# Checking Convexity of a function?

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.

• 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. Jan 7 at 16:16
• @HansEngler yes, alpha and all other parameters are assumed fixed. Jan 7 at 16:59

## 1 Answer

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.