Not an answer; just an illustration.
I doubt this helps, but here is an indication of what the polyhedron would look like in $\mathbb{R}^3$:
![PolytopeMink][1]
The origin is green, $\{A_1, A_2, A_3\}$ are the three corners of the triangle, and the ribs of $\mathrm{cone}(e_1,e_2,e_3)$ are depicted at each vertex. One then has to imagine extending the ribs to $\infty$ and taking the convex hull.