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: set_random_seed()
sage: K = random_cone(max_ambient_dim=8, max_rays=10)
sage: x = K.random_element()
sage: S = [x]
and ``G`` in the face lattice is equal to the face generated by
``F + G`` (in the Minkowski sense) [Tam]_::
- sage: set_random_seed()
sage: K = random_cone(max_ambient_dim=8, max_rays=10)
sage: L = K.face_lattice()
sage: F = L.random_element()
Combining Proposition 3.1 and Corollary 3.9 in [Tam]_ gives the
following equality for any ``y`` in ``K``::
- sage: set_random_seed()
sage: K = random_cone(max_ambient_dim=8, max_rays=10)
sage: y = K.random_element()
sage: S = [y]
"""
face_lattice = K.face_lattice()
- 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.
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.
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: 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,
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]
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()])
+ sage: all(check_prop(K,F) for F in K.face_lattice())
True
"""
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())
+ span_F = Cone((c*g for c in [1,-1] for g in F), F.lattice())
return K.dual().intersection(span_F.dual())