+ return V(sum(scaled_gens))
+
+
+def pointed_decomposition(K):
+ """
+ Every convex cone is the direct sum of a pointed cone and a linear
+ subspace. Return a pair ``(P,S)`` of cones such that ``P`` is
+ pointed, ``S`` is a subspace, and ``K`` is the direct sum of ``P``
+ and ``S``.
+
+ OUTPUT:
+
+ An ordered pair ``(P,S)`` of closed convex polyhedral cones where
+ ``P`` is pointed, ``S`` is a subspace, and ``K`` is the direct sum
+ of ``P`` and ``S``.
+
+ TESTS:
+
+ A random point in the cone should belong to either the pointed
+ subcone ``P`` or the subspace ``S``. If the point is nonzero, it
+ should lie in one but not both of them::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=8)
+ sage: (P,S) = pointed_decomposition(K)
+ sage: x = random_element(K)
+ sage: P.contains(x) or S.contains(x)
+ True
+ sage: x.is_zero() or (P.contains(x) != S.contains(x))
+ True
+ """
+ linspace_gens = [ copy(b) for b in K.linear_subspace().basis() ]
+ linspace_gens += [ -b for b in linspace_gens ]