+ 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
+
+ ::