+def basically_the_same(K1,K2):
+ r"""
+ ``True`` if ``K1`` and ``K2`` are basically the same, and ``False``
+ otherwise.
+ """
+ if K1.lattice_dim() != K2.lattice_dim():
+ return False
+
+ if K1.nrays() != K2.nrays():
+ return False
+
+ if K1.dim() != K2.dim():
+ return False
+
+ if lineality(K1) != lineality(K2):
+ return False
+
+ if K1.is_solid() != K2.is_solid():
+ return False
+
+ if K1.is_strictly_convex() != K2.is_strictly_convex():
+ return False
+
+ if len(LL(K1)) != len(LL(K2)):
+ return False
+
+ C_of_K1 = discrete_complementarity_set(K1)
+ C_of_K2 = discrete_complementarity_set(K2)
+ if len(C_of_K1) != len(C_of_K2):
+ return False
+
+ if len(K1.facets()) != len(K2.facets()):
+ return False
+
+ return True
+
+
+
+def iso_space(K):
+ r"""
+ Construct the space `W \times W^{\perp}` isomorphic to the ambient space
+ of ``K`` where `W` is equal to the span of ``K``.
+ """
+ V = K.lattice().vector_space()
+
+ # Create the space W \times W^{\perp} isomorphic to V.
+ # First we get an orthogonal (but not normal) basis...
+ M = matrix(V.base_field(), K.rays())
+ W_basis,_ = M.gram_schmidt()
+
+ W = V.subspace_with_basis(W_basis)
+ W_perp = W.complement()
+
+ return W.cartesian_product(W_perp)
+
+
+def ips_iso(K):
+ r"""
+ Construct the IPS isomorphism and its inverse from our paper.
+
+ Given a cone ``K``, the returned isomorphism will split its ambient
+ vector space `V` into a cartesian product `W \times W^{\perp}` where
+ `W` equals the span of ``K``.
+ """
+ V = K.lattice().vector_space()
+ V_iso = iso_space(K)
+ (W, W_perp) = V_iso.cartesian_factors()
+
+ # A space equivalent to V, but using our basis.
+ V_user = V.subspace_with_basis( W.basis() + W_perp.basis() )
+
+ def phi(v):
+ # Write v in terms of our custom basis, where the first dim(W)
+ # coordinates are for the W-part of the basis.
+ cs = V_user.coordinates(v)
+
+ w1 = sum([ V_user.basis()[idx]*cs[idx]
+ for idx in range(0, W.dimension()) ])
+ w2 = sum([ V_user.basis()[idx]*cs[idx]
+ for idx in range(W.dimension(), V.dimension()) ])
+
+ return V_iso( (w1, w2) )
+
+
+ def phi_inv( pair ):
+ # Crash if the arguments are in the wrong spaces.
+ V_iso(pair)
+
+ #w = sum([ sub_w[idx]*W.basis()[idx] for idx in range(0,m) ])
+ #w_prime = sum([ sub_w_prime[idx]*W_perp.basis()[idx]
+ # for idx in range(0,n-m) ])
+
+ return sum( pair.cartesian_factors() )
+
+
+ return (phi,phi_inv)
+
+
+
+def unrestrict_span(K, K2=None):
+ if K2 is None:
+ K2 = K
+
+ _,phi_inv = ips_iso(K2)
+ V_iso = iso_space(K2)
+ (W, W_perp) = V_iso.cartesian_factors()
+
+ rays = []
+ for r in K.rays():
+ w = sum([ r[idx]*W.basis()[idx] for idx in range(0,len(r)) ])
+ pair = V_iso( (w, W_perp.zero()) )
+ rays.append( phi_inv(pair) )
+
+ L = ToricLattice(W.dimension() + W_perp.dimension())
+
+ return Cone(rays, lattice=L)
+
+
+
+def restrict_span(K, K2=None):
+ r"""
+ Restrict ``K`` into its own span, or the span of another cone.
+
+ INPUT:
+
+ - ``K2`` -- another cone whose lattice has the same rank as this cone.
+
+ OUTPUT:
+
+ A new cone in a sublattice.
+
+ EXAMPLES::
+
+ sage: K = Cone([(1,)])
+ sage: restrict_span(K) == K
+ True
+
+ sage: K2 = Cone([(1,0)])
+ sage: restrict_span(K2).rays()
+ N(1)
+ in 1-d lattice N
+ sage: K3 = Cone([(1,0,0)])
+ sage: restrict_span(K3).rays()
+ N(1)
+ in 1-d lattice N
+ sage: restrict_span(K2) == restrict_span(K3)
+ True
+
+ TESTS:
+
+ The projected cone should always be solid::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8)
+ sage: K_S = restrict_span(K)
+ sage: K_S.is_solid()
+ True
+
+ And the resulting cone should live in a space having the same
+ dimension as the space we restricted it to::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8)
+ sage: K_S = restrict_span(K, K.dual() )
+ sage: K_S.lattice_dim() == K.dual().dim()
+ True
+
+ This function has ``unrestrict_span()`` as its inverse::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8, solid=True)
+ sage: J = restrict_span(K)
+ sage: K == unrestrict_span(J,K)
+ True
+
+ This function should not affect the dimension of a cone::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8)
+ sage: K.dim() == restrict_span(K).dim()
+ True
+
+ Nor should it affect the lineality of a cone::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8)
+ sage: lineality(K) == lineality(restrict_span(K))
+ True
+
+ No matter which space we restrict to, the lineality should not
+ increase::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8)
+ sage: lineality(K) >= lineality(restrict_span(K))
+ True
+ sage: lineality(K) >= lineality(restrict_span(K, K.dual()))
+ True
+
+ If we do this according to our paper, then the result is proper::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8, strictly_convex=False, solid=False)
+ sage: K_S = restrict_span(K)
+ sage: P = restrict_span(K_S.dual()).dual()
+ sage: P.is_proper()
+ True
+ sage: P = restrict_span(K_S, K_S.dual())
+ sage: P.is_proper()
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=False)
+ sage: K_S = restrict_span(K)
+ sage: P = restrict_span(K_S.dual()).dual()
+ sage: P.is_proper()
+ True
+ sage: P = restrict_span(K_S, K_S.dual())
+ sage: P.is_proper()
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8, strictly_convex=False, solid=True)
+ sage: K_S = restrict_span(K)
+ sage: P = restrict_span(K_S.dual()).dual()
+ sage: P.is_proper()
+ True
+ sage: P = restrict_span(K_S, K_S.dual())
+ sage: P.is_proper()
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=True)
+ sage: K_S = restrict_span(K)
+ sage: P = restrict_span(K_S.dual()).dual()
+ sage: P.is_proper()
+ True
+ sage: P = restrict_span(K_S, K_S.dual())
+ sage: P.is_proper()
+ True
+
+ Test the proposition in our paper concerning the duals, where the
+ subspace `W` is the span of `K^{*}`::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8, solid=False, strictly_convex=False)
+ sage: K_W = restrict_span(K, K.dual())
+ sage: K_star_W_star = restrict_span(K.dual()).dual()
+ sage: basically_the_same(K_W, K_star_W_star)
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8, solid=True, strictly_convex=False)
+ sage: K_W = restrict_span(K, K.dual())
+ sage: K_star_W_star = restrict_span(K.dual()).dual()
+ sage: basically_the_same(K_W, K_star_W_star)
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8, solid=False, strictly_convex=True)
+ sage: K_W = restrict_span(K, K.dual())
+ sage: K_star_W_star = restrict_span(K.dual()).dual()
+ sage: basically_the_same(K_W, K_star_W_star)
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8, solid=True, strictly_convex=True)
+ sage: K_W = restrict_span(K, K.dual())
+ sage: K_star_W_star = restrict_span(K.dual()).dual()
+ sage: basically_the_same(K_W, K_star_W_star)
+ True
+
+ """
+ if K2 is None:
+ K2 = K
+
+ phi,_ = ips_iso(K2)
+ (W, W_perp) = iso_space(K2).cartesian_factors()
+
+ ray_pairs = [ phi(r) for r in K.rays() ]
+
+ # Shouldn't matter?
+ #
+ #if any([ w2 != W_perp.zero() for (_, w2) in ray_pairs ]):
+ # msg = 'Cone has nonzero components in W-perp!'
+ # raise ValueError(msg)
+
+ # Represent the cone in terms of a basis for W, i.e. with smaller
+ # vectors.
+ ws = [ W.coordinate_vector(w1) for (w1, _) in ray_pairs ]
+
+ L = ToricLattice(W.dimension())
+
+ return Cone(ws, lattice=L)
+
+
+
+def lineality(K):
+ r"""
+ Compute the lineality of this cone.
+
+ The lineality of a cone is the dimension of the largest linear
+ subspace contained in that cone.
+
+ OUTPUT:
+
+ A nonnegative integer; the dimension of the largest subspace
+ contained within this cone.
+
+ REFERENCES:
+
+ .. [Rockafellar] R.T. Rockafellar. Convex Analysis. Princeton
+ University Press, Princeton, 1970.
+
+ EXAMPLES:
+
+ The lineality of the nonnegative orthant is zero, since it clearly
+ contains no lines::
+
+ sage: K = Cone([(1,0,0), (0,1,0), (0,0,1)])
+ sage: lineality(K)
+ 0
+
+ However, if we add another ray so that the entire `x`-axis belongs
+ to the cone, then the resulting cone will have lineality one::
+
+ sage: K = Cone([(1,0,0), (-1,0,0), (0,1,0), (0,0,1)])
+ sage: lineality(K)
+ 1
+
+ If our cone is all of `\mathbb{R}^{2}`, then its lineality is equal
+ to the dimension of the ambient space (i.e. two)::
+
+ sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)])
+ sage: lineality(K)
+ 2
+
+ Per the definition, the lineality of the trivial cone in a trivial
+ space is zero::
+
+ sage: K = Cone([], lattice=ToricLattice(0))
+ sage: lineality(K)
+ 0
+
+ TESTS:
+
+ The lineality of a cone should be an integer between zero and the
+ dimension of the ambient space, inclusive::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8)
+ sage: l = lineality(K)
+ sage: l in ZZ
+ True
+ sage: (0 <= l) and (l <= K.lattice_dim())
+ True
+
+ A strictly convex cone should have lineality zero::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8, strictly_convex = True)
+ sage: lineality(K)
+ 0
+
+ """
+ return K.linear_subspace().dimension()
+
+
+def codim(K):
+ r"""
+ Compute the codimension of this cone.
+
+ The codimension of a cone is the dimension of the space of all
+ elements perpendicular to every element of the cone. In other words,
+ the codimension is the difference between the dimension of the
+ ambient space and the dimension of the cone itself.
+
+ OUTPUT:
+
+ A nonnegative integer representing the dimension of the space of all
+ elements perpendicular to this cone.
+
+ .. seealso::
+
+ :meth:`dim`, :meth:`lattice_dim`
+
+ EXAMPLES:
+
+ The codimension of the nonnegative orthant is zero, since the span of
+ its generators equals the entire ambient space::
+
+ sage: K = Cone([(1,0,0), (0,1,0), (0,0,1)])
+ sage: codim(K)
+ 0
+
+ However, if we remove a ray so that the entire cone is contained
+ within the `x-y`-plane, then the resulting cone will have
+ codimension one, because the `z`-axis is perpendicular to every
+ element of the cone::
+
+ sage: K = Cone([(1,0,0), (0,1,0)])
+ sage: codim(K)
+ 1
+
+ If our cone is all of `\mathbb{R}^{2}`, then its codimension is zero::
+
+ sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)])
+ sage: codim(K)
+ 0
+
+ And if the cone is trivial in any space, then its codimension is
+ equal to the dimension of the ambient space::
+
+ sage: K = Cone([], lattice=ToricLattice(0))
+ sage: K.lattice_dim()
+ 0
+ sage: codim(K)
+ 0
+
+ sage: K = Cone([(0,)])
+ sage: K.lattice_dim()
+ 1
+ sage: codim(K)
+ 1
+
+ sage: K = Cone([(0,0)])
+ sage: K.lattice_dim()
+ 2
+ sage: codim(K)
+ 2
+
+ TESTS:
+
+ The codimension of a cone should be an integer between zero and
+ the dimension of the ambient space, inclusive::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8)
+ sage: c = codim(K)
+ sage: c in ZZ
+ True
+ sage: (0 <= c) and (c <= K.lattice_dim())
+ True
+
+ A solid cone should have codimension zero::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8, solid = True)
+ sage: codim(K)
+ 0
+
+ The codimension of a cone is equal to the lineality of its dual::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8, solid = True)
+ sage: codim(K) == lineality(K.dual())
+ True
+
+ """
+ return (K.lattice_dim() - K.dim())
+
+