from sage.all import *
+
def face_generated_by(K,S):
r"""
Return the intersection of all faces of ``K`` that contain ``S``.
The face generated by <whatever> should be a face::
- sage: set_random_seed()
sage: K = random_cone(max_ambient_dim=8, max_rays=10)
- sage: S = [K.random_element() for idx in range(0,5)]
+ sage: S = ( K.random_element() for idx in range(5) )
sage: F = face_generated_by(K, S)
sage: F.is_face_of(K)
True
The face generated by a set should always contain that set::
- sage: set_random_seed()
sage: K = random_cone(max_ambient_dim=8, max_rays=10)
- sage: S = [K.random_element() for idx in range(0,5)]
+ sage: S = ( K.random_element() for idx in range(5) )
sage: F = face_generated_by(K, S)
- sage: all([F.contains(x) for x in S])
+ sage: all(F.contains(x) for x in S)
True
The generators of a proper cone are all extreme vectors of the cone,
and therefore generate their own faces::
- sage: set_random_seed()
sage: K = random_cone(max_ambient_dim=8,
....: max_rays=10,
....: strictly_convex=True,
....: solid=True)
- sage: all([face_generated_by(K, [r]) == Cone([r]) for r in K])
+ sage: all(face_generated_by(K, [r]) == Cone([r]) for r in K)
+ True
+
+ For any point ``x`` in ``K`` and any face ``F`` of ``K``, we have
+ that ``x`` is in the relative interior of ``F`` if and only if
+ ``F`` is the face generated by ``x`` [Tam]_::
+
+ sage: K = random_cone(max_ambient_dim=8, max_rays=10)
+ sage: x = K.random_element()
+ sage: S = [x]
+ sage: F = K.face_lattice().random_element()
+ sage: expected = F.relative_interior_contains(x)
+ sage: actual = (F == face_generated_by(K, S))
+ sage: actual == expected
+ True
+
+ If ``F`` and ``G`` are two faces of ``K``, then the join of ``F``
+ and ``G`` in the face lattice is equal to the face generated by
+ ``F + G`` (in the Minkowski sense) [Tam]_::
+
+ sage: K = random_cone(max_ambient_dim=8, max_rays=10)
+ sage: L = K.face_lattice()
+ sage: F = L.random_element()
+ sage: G = L.random_element()
+ sage: expected = L.join(F,G)
+ sage: actual = face_generated_by(K, F.rays() + G.rays())
+ sage: actual == expected
+ True
+
+ Combining Proposition 3.1 and Corollary 3.9 in [Tam]_ gives the
+ following equality for any ``y`` in ``K``::
+
+ 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: actual = phi_y.span(QQ)
+ sage: expected = points_cone.linear_subspace()
+ sage: actual == expected
True
"""
face_lattice = K.face_lattice()
-
- # A list comprehension doesn't work and don't ask me why.
- candidates = [F for F in face_lattice if all([F.contains(x) for x in S])]
+ candidates = [F for F in face_lattice if all(F.contains(x) for x in S)]
# K itself is a face of K, so unless we were given a set S that
# isn't a subset of K, the candidates list will be nonempty.
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: 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: 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: 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: 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())
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