Not an answer; just an illustration.

I doubt this helps, but here is an indication of what the polyhedron would look like
in $\mathbb{R}^3$:
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
![PolytopeMink][1]
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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.


  [1]: https://i.sstatic.net/dVe5a.jpg