Wrong f-vector for unbounded polyhedra

[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.

Depends on #28625

CC: @jplab @mo271

Component: geometry

Author: Jonathan Kliem

Branch/Commit: 1c5378c

Reviewer: Jean-Philippe Labbé

Issue created by migration from https://trac.sagemath.org/ticket/26922

[kliem]

comment:1

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]

comment:2

Retarging tickets optimistically to the next milestone. If you are responsible for this ticket (either its reporter or owner) and don't believe you are likely to complete this ticket before the next release (8.7) please retarget this ticket's milestone to sage-pending or sage-wishlist.

[embray]

comment:4

As the Sage-8.8 release milestone is pending, we should delete the sage-8.8 milestone for tickets that are not actively being worked on or that still require significant work to move forward. If you feel that this ticket should be included in the next Sage release at the soonest please set its milestone to the next release milestone (sage-8.9).

[jplab]

comment:5

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.

[kliem]

comment:6

#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 @jplab:

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.

[jplab]

comment:7

ping!

[kliem]

Branch: public/26922

[kliem]

comment:8

#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

[kliem]

Author: Jonathan Kliem

[kliem]

Commit: 3c9085e

[kliem]

Dependencies: #28625

[jplab]

comment:9

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

[sagetrac-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

[sagetrac-git]

Changed commit from 3c9085e to cbcc4c4

[jplab]

comment:12

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.

[jplab]

Reviewer: Jean-Philippe Labbé

[sagetrac-git]

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

1c5378c

clearified warnings

[sagetrac-git]

Changed commit from cbcc4c4 to 1c5378c

[jplab]

comment:15

Looks good to me now

[vbraun]

Changed branch from public/26922 to 1c5378c