+
+def _rho(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: _rho(K) == K
+ True
+
+ sage: K2 = Cone([(1,0)])
+ sage: _rho(K2).rays()
+ N(1)
+ in 1-d lattice N
+ sage: K3 = Cone([(1,0,0)])
+ sage: _rho(K3).rays()
+ N(1)
+ in 1-d lattice N
+ sage: _rho(K2) == _rho(K3)
+ True
+
+ TESTS:
+
+ The projected cone should always be solid::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8)
+ sage: K_S = _rho(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 = _rho(K, K.dual() )
+ sage: K_S.lattice_dim() == K.dual().dim()
+ 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() == _rho(K).dim()
+ True
+
+ Nor should it affect the lineality of a cone::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8)
+ sage: K.lineality() == _rho(K).lineality()
+ 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: K.lineality() >= _rho(K).lineality()
+ True
+ sage: K.lineality() >= _rho(K, K.dual()).lineality()
+ 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 = _rho(K)
+ sage: K_SP = _rho(K_S.dual()).dual()
+ sage: K_SP.is_proper()
+ True
+ sage: K_SP = _rho(K_S, K_S.dual())
+ sage: K_SP.is_proper()
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=False)
+ sage: K_S = _rho(K)
+ sage: K_SP = _rho(K_S.dual()).dual()
+ sage: K_SP.is_proper()
+ True
+ sage: K_SP = _rho(K_S, K_S.dual())
+ sage: K_SP.is_proper()
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8, strictly_convex=False, solid=True)
+ sage: K_S = _rho(K)
+ sage: K_SP = _rho(K_S.dual()).dual()
+ sage: K_SP.is_proper()
+ True
+ sage: K_SP = _rho(K_S, K_S.dual())
+ sage: K_SP.is_proper()
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=True)
+ sage: K_S = _rho(K)
+ sage: K_SP = _rho(K_S.dual()).dual()
+ sage: K_SP.is_proper()
+ True
+ sage: K_SP = _rho(K_S, K_S.dual())
+ sage: K_SP.is_proper()
+ True
+
+ Test Proposition 7 in our paper concerning the duals and
+ restrictions. Generate a random cone, then create a subcone of
+ it. The operation of dual-taking should then commute with rho::
+
+ sage: set_random_seed()
+ sage: J = random_cone(max_dim = 8, solid=False, strictly_convex=False)
+ sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
+ sage: K_W_star = _rho(K, J).dual()
+ sage: K_star_W = _rho(K.dual(), J)
+ sage: _basically_the_same(K_W_star, K_star_W)
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: J = random_cone(max_dim = 8, solid=True, strictly_convex=False)
+ sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
+ sage: K_W_star = _rho(K, J).dual()
+ sage: K_star_W = _rho(K.dual(), J)
+ sage: _basically_the_same(K_W_star, K_star_W)
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: J = random_cone(max_dim = 8, solid=False, strictly_convex=True)
+ sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
+ sage: K_W_star = _rho(K, J).dual()
+ sage: K_star_W = _rho(K.dual(), J)
+ sage: _basically_the_same(K_W_star, K_star_W)
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: J = random_cone(max_dim = 8, solid=True, strictly_convex=True)
+ sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
+ sage: K_W_star = _rho(K, J).dual()
+ sage: K_star_W = _rho(K.dual(), J)
+ sage: _basically_the_same(K_W_star, K_star_W)
+ True
+
+ """
+ if K2 is None:
+ K2 = K
+
+ # First we project K onto the span of K2. This will explode if the
+ # rank of ``K2.lattice()`` doesn't match ours.
+ span_K2 = Cone(K2.rays() + (-K2).rays(), lattice=K.lattice())
+ K = K.intersection(span_K2)
+
+ # Cheat a little to get the subspace span(K2). The paper uses the
+ # rays of K2 as a basis, but everything is invariant under linear
+ # isomorphism (i.e. a change of basis), and this is a little
+ # faster.
+ W = span_K2.linear_subspace()
+
+ # We've already intersected K with the span of K2, so every
+ # generator of K should belong to W now.
+ W_rays = [ W.coordinate_vector(r) for r in K.rays() ]
+
+ L = ToricLattice(K2.dim())
+ return Cone(W_rays, lattice=L)