Let us say that you have a large (large compared to the dimension) number of hyperplanes which all intersect at a point.
These planes produce a hyper-cone in whatever dimension we are working with. The issue is trying to find the edges of this hyper cone, which is a strict subset of all lines produced by intersecting $d-1$ of these planes ($d$ being the dimension).
The simplest example is in 3D. Imagine 4 planes intersecting so that they form a pyramid.
Let us define one face as $p_1$ and counterclockwise name the other ones $p_2, p_3, p_4$.
In this case the following integer pairs define valid lines $(p_1, p_2)$; $(p_2, p_3)$; $(p_3, p_4)$; $(p_4, p_1)$.
But for example $(p_3, p_1)$ is not valid and neither would be $(p_2, p_4)$. Because the lines they intersect at are not edges of the polytope.
So in arbitrary dimensions, how do I figure out which tuples of $d-1$ elements form valid edges of the polytope?