How to find $\beta$

Question

so my problem is I have to maximize $\bar{p}^Tx$,subject to $\phi(\frac{\alpha - \bar{p}^Tx}{\sqrt{x^{T}\Sigma x}})\leq \beta$, where $\phi()$ is cumulative distribution function. I have to find $\beta$ such that the task is linear/convex/not convex. Can you please help me finding such $\beta$s?
This is the optimization:
$max \ \bar{p}^Tx$
$1^T x=1$
$\phi(\frac{\alpha - \bar{p}^Tx}{\sqrt{x^{T}\Sigma x}})\leq \beta$
Thanks for any help

Edit:
so $p \ \sim N(\bar{p},\Sigma)$, then $\bar{r}=\bar{p}^Tx$, then $\sigma^2=x^T \Sigma x$ and $r \sim N(\bar{r},\sigma^2)$
the maximization is
$max \ \bar{p}^Tx$
$1^Tx=1$
$x^T \Sigma x \leq \sigma^2$
the alternative task is
$max \ \bar{p}^Tx$
$1^Tx=1$
$P(r\leq \alpha)\leq \beta$
where $\alpha$ is known and $\beta$ is maximal probability.
the constraint $P(r\leq \alpha)\leq \beta$ is equivalent to $P(\frac{r-\bar{r}}{\sigma}\leq \frac{\alpha-\bar{r}}{\sigma}) \leq \beta$, while $\frac{r-\bar{r}}{\sigma} \sim N(0,1)$. Now we can use cumulative distribution function: $P(\frac{r-\bar{r}}{\sigma}\leq \frac{\alpha-\bar{r}}{\sigma}) = \Phi(\frac{\alpha - \bar{r}}{\sigma})$.
Alright so this above is my assignment and now my job is to find for what $\beta$ is this task linear/convex/non convex. I also had to adjust my $\Phi(\frac{\alpha - \bar{r}}{\sigma})$ to $\Phi(\frac{\alpha - \bar{p}^Tx}{\sqrt{x^T \Sigma x}})$. I hope now i gave you informations.

Answer 1

Consider the inequality $$\Phi\bigg(\frac{\alpha - \bar{p}^Tx}{\sqrt{x^T \Sigma x}}\bigg)\leq \beta \iff \frac{\alpha - \bar{p}^Tx}{\sqrt{x^T \Sigma x}} \leq \Phi^{-1}(\beta)$$ The above inequality can be equivalently written as $$ \alpha - \Phi^{-1}(\beta)\sqrt{x^T \Sigma x}-\bar{p}^Tx \leq 0$$ As $\Phi^{-1}(\beta)\leq 0$ for $\beta<0.5$ (see https://en.wikipedia.org/wiki/Quantile_function), thus, the term $- \Phi^{-1}(\beta)\sqrt{x^T \Sigma x}$ is convex. Also, the term $\alpha-\bar{p}^Tx$ is convex.
The sum of convex functions is convex, therefore, the problem becomes convex for $\beta<0.5$.
I think the problem can never become linear because of the norm term.