PolyhedronFace.affine_tangent_cone returning the polyhedron for the empty face

[louisng114]

When using the empty face for #29811, it should be undefined, but instead, it returns the original polyhedron.

Example:

sage: P = polytopes.hypercube(2)
sage: P.Hrepresentation()
(An inequality (-1, 0) x + 1 >= 0,
 An inequality (0, -1) x + 1 >= 0,
 An inequality (1, 0) x + 1 >= 0,
 An inequality (0, 1) x + 1 >= 0)
sage: P.face_lattice()[0]
A -1-dimensional face of a Polyhedron in ZZ^2
sage: P.face_lattice()[0].affine_tangent_cone().Hrepresentation()
(An inequality (-1, 0) x + 1 >= 0,
 An inequality (0, -1) x + 1 >= 0,
 An inequality (1, 0) x + 1 >= 0,
 An inequality (0, 1) x + 1 >= 0)

CC: @mkoeppe @jplab @kliem @yuan-zhou

Component: geometry

Author: Louis Ng, Jonathan Kliem

Branch/Commit: 4340762

Reviewer: Matthias Koeppe

Issue created by migration from https://trac.sagemath.org/ticket/32658

[kliem]

Commit: 73a4ca6

[kliem]

Branch: public/32658

[kliem]

comment:3

I do not agree that it should be undefined. It should just be empty. This is also what the documentation claims and I think the definition makes sense:

    It is equal to the sum of ``self`` and the cone of feasible directions
    at any point of the relative interior of ``self``.

---

New commits:

73a4ca6

the affine tangent cone of the empty face should be empty

[kliem]

Changed author from Louis Ng to Louis Ng, Jonathan Kliem

[mkoeppe]

comment:4

But can an "affine cone" be empty? An ordinary cone cannot - it always contains 0

[mkoeppe]

comment:5

Replying to @kliem:

It is equal to the sum of self and the cone of feasible directions
at any point of the relative interior of self.

I would say that this sentence does not define anything for the empty case, because "the" cone at "any" point of the empty set is undefined.

[sagetrac-git]

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

4340762

throw an error, because it is not a cone

[sagetrac-git]

Changed commit from 73a4ca6 to 4340762

[kliem]

New commits:

4340762

throw an error, because it is not a cone

[kliem]

comment:8

Replying to @mkoeppe:

Replying to @kliem:

It is equal to the sum of self and the cone of feasible directions
at any point of the relative interior of self.

I would say that this sentence does not define anything for the empty case, because "the" cone at "any" point of the empty set is undefined.

Agreed.

[mkoeppe]

Reviewer: Matthias Koeppe

[kliem]

comment:10

Thank you.

[vbraun]

Changed branch from public/32658 to 4340762