Wrong f-vector for unbounded polyhedra

[gh-kliem]

#28625 fixed the f_vector in the case of unpointed polyhedra/polyhedra with lines.

We add doctests showing that #28625 fixed a bug in f_vector.

Before:

sage: Polyhedron(ieqs=[[1,-1,0],[1,1,0]]).f_vector()
(1, 2, 1)

But this polyhedron does not have zero-dimensional faces, and #28625 has correctly changed that:

sage: Polyhedron(ieqs=[[1,-1,0],[1,1,0]]).f_vector()
(1, 0, 2, 1)

Also we add documentation, specifically warning users that the methods

• vertices,
• vertices_list,
• vertices_generator
• vertices_matrix,
• n_vertices,

treat vertices of the Vrepresentation and not vertices of the polyhedron in the unpointed case.

[gh-kliem]

This will be probably solved at some point by #26887 The calculation is correct there.

Also it can be corrected quickly now by calculating the dimension of the first level in the face lattice.

[embray]

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[embray]

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[jipilab]

In principle, the problem is that the face lattice has four elements, but the elements on the second level are not 0-dimensional, but 1-dimensional.

So, depending on the definition of the f-vector:

• Counting the number of elements at each level of the face lattice
• Vector counting the number of faces by dimension

It should return different things.

One thing that should be done, is to make this explicit in the documentation of f_vector once this ticket is fixed and give an example of the difference with unbounded polyhedra.

[gh-kliem]

#27063 will change the calculation of the f-vector (hopefully soon) to make use of CombinatorialPolyhedron.

This will count the number of faces per dimension, as the docstring of f_vector states (there is a +2 or -2 two missing, as the first entry gives the -1-dimensional faces count.

We could make this ticket depend on #27063 and then just append the docstring accordingly.

One could also fix it for now (just append a few zeros according to n_lines).

Replying to jipilab:

In principle, the problem is that the face lattice has four elements, but the elements on the second level are not 0-dimensional, but 1-dimensional.

So, depending on the definition of the f-vector:

• Counting the number of elements at each level of the face lattice

I think f-vectors of fixed dimension polyhedra should be have sums/addition. Also if one considers how adding or deleting inequalities can alter the f-vector this is strange. Anyway, it's most important to be precise, which one we use. Adding a zero or deleting it, if one really depends one version is trivial.

• Vector counting the number of faces by dimension

It should return different things.

One thing that should be done, is to make this explicit in the documentation of f_vector once this ticket is fixed and give an example of the difference with unbounded polyhedra.

[jipilab]

ping!

[gh-kliem]

#28625 fixes this.

I added some tests, improved the documentation of f_vector and fixed a tiny typo.

---

New commits:

​b89610e	added combinatorial polyhedron as an attribute for polyhedra
​326602c	f_vector of CombinatorialPolyhedron is a vector
​dfbe2ad	Merge branch 'public/28607' of git://trac.sagemath.org/sage into public/28621
​ed5518b	used CombinatorialPolyhedron to compute f_vector
​9bdd005	give an error message for polytopes in some cases; removed incorrect example
​acd671d	now we get a precice error message for inexact truncated dodecahedron
​bf85a62	subsequent calls for f_vector fail if first attempt fails
​3c9085e	give doctests that f_vector for unbounded polyhedra works now

[jipilab]

It would be nice if the note that you gave be in a NOTE environment followed by an appropriate example illustrating what is meant.

[git]

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

​cbcc4c4

Comment to f_vector in NOTE environment; warnings of ambigious meaning of vertex

[jipilab]

May I suggest the following:

-            In case of a polyhedron with lines (unpointed polyhedron),
-            return the number of vertices of the ``Vrepresentation``.
-            Wheras the polyhedron has no vertices, this number corresponds
-            to the number of `k`-faces, where `k` is the number of lines.
+            If the polyhedron has lines, returns the number of vertices in
+            the ``Vrepresentation``. As the represented polyhedron has 
+            no 0-dimensional faces (i.e. vertices), this number corresponds
+            to the number of `k`-faces, where `k` is the number of lines.

Somehow, even though I corrected the above warning. I do not like what it says at all for the following reason:

sage: P = Polyhedron(rays=[[1,0,0]],lines=[[0,1,0]])
sage: P.vertices()
(A vertex at (0, 0, 0),)
sage: Q = Polyhedron(lines=[[0,1,0],[0,0,1]])
sage: Q.vertices()
(A vertex at (0, 0, 0),)
sage: R = Polyhedron(rays=[[1,0,0],[0,1,0]],vertices=[[0,1,0],[1,0,0]],lines=[[0,0,1]])
sage: R.vertices()
(A vertex at (0, 1, 0), A vertex at (1, 0, 0))

In Sage, the computational convention (or compromise) is that polyhedra without vertices in their V-representation still receive a canonically computed vertex in order to do computations.

With this in mind, the second sentence is wrong. Just look at the above polyhedron R. It has lines, and two vertices.

---> This warning is more confusing than anything. Please rephrase taking the above three examples in mind and the convention in Sage.

[git]

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

​1c5378c

clearified warnings

[jipilab]

Looks good to me now