Trac #29323: Set up cross polytope with both Vrep and Hrep (if backen…

…d supports it)

We set up the cross polytope with precomputed double description, if the
backend supports it. Otherwise, the vertices will be used as before (the
shorter input is selected automatically by #28880).

Before this ticket:

{{{
sage: %time P = polytopes.cross_polytope(8, backend='field')
CPU times: user 14.3 s, sys: 4.01 ms, total: 14.3 s
Wall time: 14.3 s
sage: %timeit P = polytopes.cross_polytope(13, backend='ppl')
1 loop, best of 5: 312 ms per loop
}}}

With this ticket:

{{{
sage: %time P = polytopes.hypercube(8, backend='field')
CPU times: user 29.8 ms, sys: 11.2 ms, total: 41 ms
Wall time: 40.3 ms
sage: %timeit P = polytopes.cross_polytope(13, backend='ppl')
1 loop, best of 5: 312 ms per loop

sage: %timeit P = polytopes.hypercube(13, backend='field')
10 loops, best of 5: 69.5 ms per loop
}}}

Note: This might slow down some backends (e.g. `ppl`) a bit, as the
inequalities are precomputed and then discarded again. However, this
seems to be only minor. The speed up for backend `field` is significant,
as nothing is being computed anymore.

URL: https://trac.sagemath.org/29323
Reported by: gh-kliem
Ticket author(s): Jonathan Kliem
Reviewer(s): Frédéric Chapoton

src/sage/geometry/polyhedron/library.py

@@ -2871,7 +2871,7 @@ def hypercube(self, dim, intervals=None, backend=None):
 
         else:
             raise ValueError("the dimension of the hypercube must match the number of intervals")
-        return parent([cp, [], []], [ieqs, []], convert=convert, Vrep_minimal=True, Hrep_minimal=True )
+        return parent([cp, [], []], [ieqs, []], convert=convert, Vrep_minimal=True, Hrep_minimal=True)
 
     def cube(self, intervals=None, backend=None):
         r"""
@@ -2960,10 +2960,19 @@ def cross_polytope(self, dim, backend=None):
 
             sage: cp = polytopes.cross_polytope(4,backend='normaliz')   # optional - pynormaliz
             sage: TestSuite(cp).run()                                   # optional - pynormaliz
+
+        Check that double description is set up correctly::
+
+            sage: P = polytopes.cross_polytope(6, backend='ppl')
+            sage: Q = polytopes.cross_polytope(6, backend='ppl')
+            sage: P == Q
+            True
         """
-        verts = list((ZZ**dim).basis())
-        verts.extend([-v for v in verts])
-        return Polyhedron(vertices=verts, backend=backend)
+        verts = tuple((ZZ**dim).basis())
+        verts += tuple(-v for v in verts)
+        ieqs = tuple((1,) + x for x in itertools.product([-1,1], repeat=dim))
+        parent = Polyhedra(ZZ, dim, backend=backend)
+        return parent([verts, [], []], [ieqs, []], Vrep_minimal=True, Hrep_minimal=True)
 
     def parallelotope(self, generators, backend=None):
         r"""