fix is_simplicial for non-full-dimensional polytopes

[gh-LaisRast]

The current is_simplicial method returns False if the polytope is not full-dimensional.

sage: P = polytopes.simplex()
sage: P
A 3-dimensional polyhedron in ZZ^4 defined as the convex hull of 4 vertices
sage: P.is_simplicial()
False
sage: P.affine_hull().is_simplicial()
True

[gh-LaisRast]

New commits:

​3087931

fix is_simplicial for non-full-dimensional polytopes

[gh-LaisRast]

On a similar topic: The method is_simple returns False when the polyhedron is not a polytope. I would suggest to fix that since simpleness is defined for polyhedra, or just raise NotImplementedError like in is_simplicial.

[gh-kliem]

I guess you could just delete if not self.is_compact(): return False in is_simple. But for non-pointed polyhedra we should raise an NotImplementedError as it is really unclear what is intended in this case.

Then it just checks if every vertex is adjacent to d facets.

Add an example which shows what works correct now (the example from the description should be fine).

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​ecc2973

add an example

[gh-LaisRast]

In any case, the current behavior of is_simple with polyhedra is not correct. This could be done in another ticket.

I added the simplex example in is_simplicial.

Replying to gh-kliem:

I guess you could just delete if not self.is_compact(): return False in is_simple. But for non-pointed polyhedra we should raise an NotImplementedError as it is really unclear what is intended in this case.

Then it just checks if every vertex is adjacent to d facets.

Add an example which shows what works correct now (the example from the description should be fine).

[vdelecroix]

Wouldn't the implementation be simpler and faster with

M = self.incidence_matrix()
for face in self.Hrepresentation():
    if facet.is_inequality() and sum(M.column(face.index()) != d:
        return False
return True

[gh-kliem]

If we use the incidence matrix anyway, than we can just restrict to using it. Let n be the number of vertices. Then a polytope is simplicial if and only if

not any(d < sum(M.column(i)) < n for i in M.ncols())

I guess the current solution was intended to change as little as possible.

[jipilab]

Looks good to me.