cone/faces.py: add the missing HilgertHofmannLawson reference that I cited.

[sage.d.git] / mjo / cone / faces.py

diff --git a/mjo/cone/faces.py b/mjo/cone/faces.py
index a6820164999e55d6396d5d1b081419e57f5c791b..70706a8ca82383c25d98a17d28a333d8d942f83a 100644 (file)
--- a/mjo/cone/faces.py
+++ b/mjo/cone/faces.py
@@ -143,13 +143,17 @@ def dual_face(K,F):
 
     REFERENCES:
 
+    .. [HilgertHofmannLawson] Joachim Hilgert, Karl Heinrich Hofmann, and Jimmie
+       D. Lawson. Lie groups, convex cones and semigroups. Oxford Mathematical
+       Monographs. Clarendon Press, Oxford, 1989. ISBN 9780198535690.
+
     .. [Tam] Bit-Shun Tam. On the duality operator of a convex cone. Linear
        Algebra and its Applications, 64:33-56, 1985, doi:10.1016/0024-3795(85)
        90265-4.
 
     SETUP::
 
-        sage: from mjo.cone.faces import dual_face
+        sage: from mjo.cone.faces import (dual_face, face_generated_by)
 
     EXAMPLES:
 
@@ -189,6 +193,29 @@ def dual_face(K,F):
         sage: dual_face(K,trivial_face).is_equivalent(K.dual())
         True
 
+    The dual of the cone of ``K`` at ``y`` is the dual face of the face
+    of ``K`` generated by ``y`` ([Tam]_ Corollary 3.2)::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_ambient_dim=8, max_rays=10)
+        sage: y = K.random_element()
+        sage: S = [y]
+        sage: phi_y = face_generated_by(K,S)
+        sage: points_cone_gens = list(K.rays()) + [-z for z in phi_y.rays()]
+        sage: points_cone = Cone(points_cone_gens, K.lattice())
+        sage: points_cone.dual().is_equivalent(dual_face(K, phi_y))
+        True
+
+    Since all faces of a polyhedral cone are exposed, the dual face of a
+    dual face should be the original face [HilgertHofmannLawson]_::
+
+        sage: set_random_seed()
+        sage: def check_prop(K,F):
+        ....:     return dual_face(K.dual(), dual_face(K,F)).is_equivalent(F)
+        sage: K = random_cone(max_ambient_dim=8, max_rays=10)
+        sage: all([check_prop(K,F) for F in K.face_lattice()])
+        True
+
     """
     # Ensure that F is actually a face of K before continuing.
     if not F.is_face_of(K):