+ If ``K`` is strictly convex, then both ``K_W`` and
+ ``K_star_W.dual()`` should equal ``K`` (after we unrestrict)::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 10, strictly_convex=True)
+ sage: K_W = restrict_span(intersect_span(K,K.dual()), K.dual())
+ sage: K_star_W_star = restrict_span(K.dual()).dual()
+ sage: j1 = unrestrict_span(K_W, K.dual())
+ sage: j2 = unrestrict_span(K_star_W_star, K.dual())
+ sage: j1 == j2
+ True
+ sage: j1 == K
+ True
+ sage: K; [ list(r) for r in K.rays() ]
+
+ 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 = 10, solid=False, strictly_convex=False)
+ sage: K_W = restrict_span(intersect_span(K,K.dual()), K.dual())
+ sage: K_star_W_star = restrict_span(K.dual(), K.dual()).dual()
+ sage: K_W.nrays() == K_star_W_star.nrays()
+ True
+ sage: K_W.dim() == K_star_W_star.dim()
+ True
+ sage: lineality(K_W) == lineality(K_star_W_star)
+ True
+ sage: K_W.is_solid() == K_star_W_star.is_solid()
+ True
+ sage: K_W.is_strictly_convex() == K_star_W_star.is_strictly_convex()
+ 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() ]
+
+ 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 = 10)
+ 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 = 10, strictly_convex = True)
+ sage: lineality(K)
+ 0
+