Containment checks are wrong for empty polyhedra

[novoselt]

sage: p = Polyhedron()
sage: p
The empty polyhedron in QQ^0.
sage: p.contains(0)
True
sage: p.contains(1)
True
sage: p.contains([0, 1])
True
sage: p.contains([(0, 1)])
True
sage: p.interior_contains([0, 1])
True
sage: p.relative_interior_contains([0, 1])
True

I think all of the above should return False.

[vbraun]

I changed the "empty" polyhedron to contain one equation 0==-1. I also modified the "contains" checks to be slightly more general and work with points whose coordinates are not necessarily in the same field as the polyhedron.

[novoselt]

Replying to vbraun:

I changed the "empty" polyhedron to contain one equation 0==-1.

This seems reasonable, but it shows in the doctest as "[An equation () x + -1 == 0]" which is confusing: what is "() x" and why "+ -"?

This is not directly related to this ticket, but how about omitting "() x +" for 0-dimensional equations/inequalities and taking into account the "+ -" situation in general?

[vbraun]

I guess this ticket is as good as any other to fix the string '_repr_()' of (in)equalities :-) The updated patch skips "zero coefficient times x" and handles the sign of the constant term nicer.

[novoselt]

Patch order issues:

applying trac_10238_containment_check_for_empty_polyhedra.patch
patching file sage/geometry/polyhedra.py
Hunk #8 succeeded at 4208 with fuzz 2 (offset 446 lines).
now at: trac_10238_containment_check_for_empty_polyhedra.patch

(Tested on top of face lattice.)

I am fine with this version, but just to make sure: you really want to start the line with "+" if there are no variables and have space between the sign and the coefficient only if there are variables?

[vbraun]

I rearranged the patches an in #9604, should apply cleanly now.

My preferred prettyprinting is

  An equation (-1) x + 1 >= 0
  An equation (2) x - 1 >= 0
  An equation 1 == 0
  An equation -1 == 0

and I fixed the 3rd one in the patch.

[vbraun]

Updated patch

[novoselt]

Cool!