+ sage: P = rho(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 = rho(K)
+ sage: P = rho(K_S.dual()).dual()
+ sage: P.is_proper()
+ True
+ sage: P = rho(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 = rho(K)
+ sage: P = rho(K_S.dual()).dual()
+ sage: P.is_proper()
+ True
+ sage: P = rho(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 = rho(K, K.dual())
+ sage: K_star_W_star = rho(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 = rho(K, K.dual())
+ sage: K_star_W_star = rho(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 = rho(K, K.dual())
+ sage: K_star_W_star = rho(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 = rho(K, K.dual())
+ sage: K_star_W_star = rho(K.dual()).dual()
+ sage: basically_the_same(K_W, K_star_W_star)
+ True
+
+ """
+ if K2 is None:
+ K2 = K
+
+ # First we project K onto the span of K2. This can be done with
+ # cones (i.e. without converting to vector spaces), but it's
+ # annoying to deal with lattice mismatches.
+ span_K2 = Cone(K2.rays() + (-K2).rays(), lattice=K.lattice())
+ K = K.intersection(span_K2)
+
+ V = K.lattice().vector_space()
+
+ # Create the space W \times W^{\perp} isomorphic to V.
+ # First we get an orthogonal (but not normal) basis...
+ W_basis = drop_dependent(K2.rays())
+ W = V.subspace_with_basis(W_basis)
+
+ # 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)
+
+
+
+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: