- sage: K = random_cone(max_dim = 10, 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: codim(K)
- 0
-
- sage: K = Cone([(0,)])
- sage: codim(K)
- 1
-
- sage: K = Cone([(0,0)])
- sage: codim(K)
- 2
-
- TESTS:
+ 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