Improvements for `is_reflexive` for polyhedra over the integers

[kliem]

Currently, this method has almost no documentation.

Also it relies on polar to work differently for Polyhedron_ZZ than for Polyhedron_base. However, this is not being tested at all. We fix this by checking being reflexive just from the inequalities.

This also improves the error messages. We raise a ValueError if the polyhedron is unbounded. If the polytope does not have the origin in the interior, we return False, as this is clearly not reflexive.

CC: @jplab @LaisRast

Component: geometry

Keywords: polytopes, is_reflexive

Author: Jonathan Kliem

Branch/Commit: 27fa75c

Reviewer: Jean-Philippe Labbé

Issue created by migration from https://trac.sagemath.org/ticket/29229

[kliem]

Branch: public/29229

[kliem]

New commits:

1d71889

some improvements for `is_reflexive`

[kliem]

Commit: 1d71889

[kliem]

Changed keywords from none to polytopes, is_reflexive

[kliem]

Description changed:

--- 
+++ 
@@ -2,4 +2,4 @@
 
 Also it relies on `polar` to work differently for `Polyhedron_ZZ` than for `Polyhedron_base`. However, this is not being tested at all. We fix this by checking being reflexive just from the inequalities.
 
-This also improves the error messages. We raise a `ValueError` is the polyhedron is unbounded. If the polytope does not have the origin in the interior, we return `False`, as this is clearly not reflexive.
+This also improves the error messages. We raise a `ValueError` if the polyhedron is unbounded. If the polytope does not have the origin in the interior, we return `False`, as this is clearly not reflexive.

[jplab]

comment:3

Some remarks:

-        A lattice polytope is reflexive if it is polar to a lattice polytope.
+        A lattice polytope is reflexive if its polar is a lattice polytope.

Even though the raised error is not wrong as written (that it is not a polytope), I would still say "the polyhedron is not compact" or "the polyhedron should be compact". Simply because this avoids an eventual confusion in nomenclature polytope vs polyhedron.

[kliem]

comment:4

Replying to @jplab:

Some remarks:

-        A lattice polytope is reflexive if it is polar to a lattice polytope.
+        A lattice polytope is reflexive if its polar is a lattice polytope.

Those two statements are not equivalent. I would say that a polytope without origin in interior is not reflexive and that the return value shall be False (and no error message). However, in this case the polar is not defined. This is why I chose this specific formulation.

Even though the raised error is not wrong as written (that it is not a polytope), I would still say "the polyhedron is not compact" or "the polyhedron should be compact". Simply because this avoids an eventual confusion in nomenclature polytope vs polyhedron.

[jplab]

comment:5

Replying to @kliem:

Replying to @jplab:

Some remarks:

-        A lattice polytope is reflexive if it is polar to a lattice polytope.
+        A lattice polytope is reflexive if its polar is a lattice polytope.

Those two statements are not equivalent. I would say that a polytope without origin in interior is not reflexive and that the return value shall be False (and no error message). However, in this case the polar is not defined. This is why I chose this specific formulation.

Then, one should be transparent. Why not something like:

+       A lattice polytope is reflexive if it contains the origin and its polar with respect to the origin is a lattice polytope.