Generate triangular surface mesh of convex hull spanned by 8 points in 3D-Space
You don't actually need the convex hull. This is a simple linear programming instance: find $a_i$ such that $$0 \le a_i \\
\sum a_i = 1 \\
\sum a_i \phi_{i, 1} \le 0 \\
\sum a_i \phi_{i, 2} = 0 \\
\sum a_i \phi_{i, 3} = 0$$ maximising $\sum a_i \phi_{i, 1}$. There are optimised LP solvers out there which you can use.