I need to determine the boundary of the following open convex cone:
$$\Omega = \left\{ (x_1, x_2, x_3, x_4, x_5) \in \mathbb R^5: x_1 > 0, x_1x_2 - x_4^2 > 0, x_1x_2x_3 - x_3x_4^2 - x_2x_5^2 > 0\right\}.$$
For a cartesian product of sets $X$ and $Y$, it is possible to use the rule $\partial(X \times Y) = (X \times \partial Y) \cup (\partial X \times Y)$. However, I do not think that helps in this case, as there is 'mixing' of the defining inequalities, since the $x_1$ term appears in each inequality.
It's possible I'm missing a/some result/s from convex analysis, or other areas I haven't looked at in a while. Any help would be appreciated, thanks.