X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Ffaces.py;h=70706a8ca82383c25d98a17d28a333d8d942f83a;hb=d1017f501eb2058908a409818f143d2376863aa6;hp=8e5f2436c47802f4e8c61fbb2166f44d7370537d;hpb=c00b0cce43e8345d88097c74a3421f71e3b45e2b;p=sage.d.git

diff --git a/mjo/cone/faces.py b/mjo/cone/faces.py
index 8e5f243..70706a8 100644
--- a/mjo/cone/faces.py
+++ b/mjo/cone/faces.py
@@ -1,5 +1,6 @@
 from sage.all import *
 
+
 def face_generated_by(K,S):
     r"""
     Return the intersection of all faces of ``K`` that contain ``S``.
@@ -130,3 +131,95 @@ def face_generated_by(K,S):
         raise ValueError('S is not a subset of the cone')
     else:
         return face_lattice.sorted(candidates)[0]
+
+
+def dual_face(K,F):
+    r"""
+    Return the dual face of ``F`` with respect to the cone ``K``.
+
+    OUTPUT:
+
+    A face of ``K.dual()``.
+
+    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, face_generated_by)
+
+    EXAMPLES:
+
+    The dual face of the first standard basis vector in three dimensions
+    is the face generated by the other two standard basis vectors::
+
+        sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)])
+        sage: F = Cone([(1,0,0)])
+        sage: dual_face(K,F).rays()
+        M(0, 0, 1),
+        M(0, 1, 0)
+        in 3-d lattice M
+
+    TESTS:
+
+    The dual face of ``K`` with respect to itself should be the
+    lineality space of its dual [Tam]_::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_ambient_dim=8, max_rays=10)
+        sage: K_dual = K.dual()
+        sage: lKd_gens = [ dir*l for dir in [1,-1] for l in K_dual.lines() ]
+        sage: linspace_K_dual = Cone(lKd_gens, K_dual.lattice())
+        sage: dual_face(K,K).is_equivalent(linspace_K_dual)
+        True
+
+    If ``K`` is proper, then the dual face of its trivial face is the
+    dual of ``K`` [Tam]_::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_ambient_dim=8,
+        ....:                 max_rays=10,
+        ....:                 strictly_convex=True,
+        ....:                 solid=True)
+        sage: L = K.lattice()
+        sage: trivial_face = Cone([L.zero()], L)
+        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):
+        raise ValueError("%s is not a face of %s" % (F,K))
+
+    span_F = Cone([c*g for c in [1,-1] for g in F], F.lattice())
+    return K.dual().intersection(span_F.dual())