Dimension of Polyhedron is not well defined

[jipilab]

Given a vector space V and a polyhedron P, the dimension of P is the dimension of its affine hull, i.e. the smallest affine subspace of V in which P can live. (For a reference, see G.M. Ziegler's book "Lectures on Polytopes" p.5)

Right now, the dimension of a Polyhedron is defined as:

    def dim(self):
        """
        Return the dimension of the polyhedron.

        EXAMPLES::

            sage: simplex = Polyhedron(vertices = [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]])
            sage: simplex.dim()
            3
            sage: simplex.ambient_dim()
            4
       """
        return self.ambient_dim() - self.n_equations()

But this is obviously not the truthful definition. With the way things are now, this leads to various problems such as:

sage: P1=Polyhedron(vertices=[[1,0,0],[0,1,0],[0,0,1]])
sage: P2=Polyhedron(vertices=[[2,0,0],[0,2,0],[0,0,2]])
sage: Inter=P1.intersection(P2)
sage: Inter
The empty polyhedron in QQ^3.
sage: Inter.dim()
1

While the right answer should be -1.

One can produce many other examples...

Apply trac_12193_fix_dimension.patch

[vbraun]

Fixed by attached patch.

[davidloeffler]

I've added a new patch which will stop the patchbot whining about trailing whitespace and commit messages.

Apply trac_12193_fix_dimension-v2.patch

[vbraun]

I'm fine with your changes :-)

[vbraun]

Rediffed for sage-5.0.beta11, nothing of interest changed.

[jdemeyer]

Please fill in your real name as Reviewer.

[vbraun]

Rebased for #11763

[vbraun]

For the patchbot: apply trac_12193_fix_dimension.patch only

[vbraun]

The patchbot found a machine-dependent issue in a cmp() doctest, fixed now.

[vbraun]

Updated patch

[vbraun]

Updated patch to push one patch hunk down to #11763.