Polyhedra, facets and vertices
I have a 6-dimensional, convex, compact polyhedron in $\mathbb{R}^{12}$, that I am calling P1.
The polyhedron P1 was specified by giving a large number of inequalities.
When I enter P1.faces(5) into Sage, I get the following
(A 5-dimensional face of a Polyhedron in RDF^12 defined as the convex hull of 6 vertices,
A 5-dimensional face of a Polyhedron in RDF^12 defined as the convex hull of 7 vertices,
A 5-dimensional face of a Polyhedron in RDF^12 defined as the convex hull of 7 vertices,
A 6-dimensional face of a Polyhedron in RDF^12 defined as the convex hull of 7 vertices,
A 5-dimensional face of a Polyhedron in RDF^12 defined as the convex hull of 6 vertices,
A 6-dimensional face of a Polyhedron in RDF^12 defined as the convex hull of 8 vertices,
...)
How can there be two 6-dimensional faces, each with a different number of vertices,
on the boundary of P1? I have checked that P1 really does have dimension 6.
Also, when I try to find a list of vertices on the boundary of any 5-dimensional face of P1,
for example
P1.faces(5)[0].vertices_list()
I get an error message
AttributeError: 'tuple' object has no attribute 'vertices_list'
However, P1.faces(5)[0].vertices() works.
My second question is, how do I get a list of vertices on the boundary of an $n$-dimensional face?
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