set up the hypercube with both Vrep and Hrep

src/sage/geometry/polyhedron/library.py

@@ -86,6 +86,7 @@
 from sage.combinat.permutation import Permutations
 from sage.groups.perm_gps.permgroup_named import AlternatingGroup
 from .constructor import Polyhedron
+from .parent import Polyhedra
 from sage.graphs.digraph import DiGraph
 from sage.combinat.root_system.associahedron import Associahedron
 
@@ -2792,6 +2793,13 @@ def hypercube(self, dim, intervals=None, backend=None):
             ...
             ValueError: the dimension of the hypercube must match the number of intervals
 
+        The intervals must be pairs `(a, b)` with `a < b`::
+
+            sage: w_cube = polytopes.hypercube(3, intervals = [[0,1],[3,2],[0,3]])
+            Traceback (most recent call last):
+            ...
+            ValueError: each interval must be a pair `(a, b)` with `a < b`
+
         If a string besides 'zero_one' is passed to ``intervals``, return an
         error::
 
@@ -2800,18 +2808,42 @@ def hypercube(self, dim, intervals=None, backend=None):
             ...
             ValueError: the only allowed string is 'zero_one'
         """
+        parent = Polyhedra(ZZ, dim, backend=backend)
+
+        # Preparing the inequalities:
+        # If the intervals are (a_1,b_1), ..., (a_dim, b_dim),
+        # then the inequalites correspond to
+        # b_1,b_2,...,b_dim, a_1,a_2,...,a_dim
+        # in that order.
+        ieqs = [[0]*(dim+1) for _ in range(2*dim)]
+        for i in range(dim):
+            ieqs[i][i+1] = -1
+            ieqs[dim+i][i+1] = 1
+
         if intervals is None:
             cp = list(itertools.product([-1,1], repeat=dim))
+            for i in range(dim):
+                ieqs[i][0]     = 1  # An inequality -x_i + 1 >= 0
+                ieqs[i+dim][0] = 1  # An inequality  x_i + 1 >= 0
         elif isinstance(intervals, str):
             if intervals == 'zero_one':
                 cp = list(itertools.product([0,1], repeat=dim))
+                for i in range(dim):
+                    ieqs[i][0] = 1  # An inequality -x_i + 1 >= 0
             else:
                 raise ValueError("the only allowed string is 'zero_one'")
         elif len(intervals) == dim:
+            if not all(a < b for a,b in intervals):
+                raise ValueError("each interval must be a pair `(a, b)` with `a < b`")
+            parent = parent.base_extend(intervals)
             cp = list(itertools.product(*intervals))
+            for i in range(dim):
+                ieqs[i][0]     =  intervals[i][1]  # An inequality -x_i + b_i >= 0
+                ieqs[i+dim][0] = -intervals[i][0]  # An inequality  x_i - a_i >= 0
+
         else:
             raise ValueError("the dimension of the hypercube must match the number of intervals")
-        return Polyhedron(vertices=cp, backend=backend)
+        return parent.element_class(parent, [cp, [], []], [ieqs, []], Vrep_minimal=True, Hrep_minimal=True )
 
     def cube(self, intervals=None, backend=None):
         r"""