Trac #22275: Implement duality of faces for reflexive polytopes

It was "secretly" implemented in the old code removed in #22122 by order
of faces and while it was not documented in Sage, it was dissipated in
some worksheets and papers.

URL: https://trac.sagemath.org/22275
Reported by: novoselt
Ticket author(s): Andrey Novoseltsev
Reviewer(s): Julian Rüth

src/sage/geometry/lattice_polytope.py

@@ -1657,6 +1657,36 @@ def distances(self, point=None):
             self._distances.set_immutable()
             return self._distances
 
+    @cached_method
+    def dual(self):
+        r"""
+        Return the dual face under face duality of polar reflexive polytopes.
+        
+        This duality extends the correspondence between vertices and facets.
+        
+        OUTPUT:
+        
+        - a :class:`lattice polytope <LatticePolytopeClass>`.
+        
+        EXAMPLES::
+        
+            sage: o = lattice_polytope.cross_polytope(4)
+            sage: e = o.edges()[0]; e
+            1-d face of 4-d reflexive polytope in 4-d lattice M
+            sage: ed = e.dual(); ed
+            2-d face of 4-d reflexive polytope in 4-d lattice N
+            sage: ed.ambient() is e.ambient().polar()
+            True
+            sage: e.ambient_vertex_indices() == ed.ambient_facet_indices()
+            True
+            sage: e.ambient_facet_indices() == ed.ambient_vertex_indices()
+            True
+        """
+        for f in self._ambient.polar().faces(codim=self.dim() + 1):
+            if f._ambient_vertex_indices == self._ambient_facet_indices:
+                f.dual.set_cache(self)
+                return f
+
     @cached_method
     def dual_lattice(self):
         r"""