-def motzkin_decomposition(K):
- r"""
- Return the pair of components in the motzkin decomposition of this cone.
-
- Every convex cone is the direct sum of a strictly convex cone and a
- linear subspace. Return a pair ``(P,S)`` of cones such that ``P`` is
- strictly convex, ``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 strictly convex, ``S`` is a subspace, and ``K`` is the
- direct sum of ``P`` and ``S``.
-
- EXAMPLES:
-
- The nonnegative orthant is strictly convex, so it is its own
- strictly convex component and its subspace component is trivial::
-
- sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)])
- sage: (P,S) = motzkin_decomposition(K)
- sage: K.is_equivalent(P)
- True
- sage: S.is_trivial()
- True
-
- Likewise, full spaces are their own subspace components::
-
- sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
- sage: K.is_full_space()
- True
- sage: (P,S) = motzkin_decomposition(K)
- sage: K.is_equivalent(S)
- True
- sage: P.is_trivial()
- True
-
- TESTS:
-
- A random point in the cone should belong to either the strictly
- convex component or the subspace component. If the point is nonzero,
- it cannot be in both::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8)
- sage: (P,S) = motzkin_decomposition(K)
- sage: x = K.random_element()
- sage: P.contains(x) or S.contains(x)
- True
- sage: x.is_zero() or (P.contains(x) != S.contains(x))
- True
-
- The strictly convex component should always be strictly convex, and
- the subspace component should always be a subspace::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8)
- sage: (P,S) = motzkin_decomposition(K)
- sage: P.is_strictly_convex()
- True
- sage: S.lineality() == S.dim()
- True
-
- The generators of the strictly convex component are obtained from
- the orthogonal projections of the original generators onto the
- orthogonal complement of the subspace component::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8)
- sage: (P,S) = motzkin_decomposition(K)
- sage: S_perp = S.linear_subspace().complement()
- sage: A = S_perp.matrix().transpose()
- sage: proj = A * (A.transpose()*A).inverse() * A.transpose()
- sage: expected = Cone([ proj*g for g in K ], K.lattice())
- sage: P.is_equivalent(expected)
- True
- """
- linspace_gens = [ copy(b) for b in K.linear_subspace().basis() ]
- linspace_gens += [ -b for b in linspace_gens ]
-
- S = Cone(linspace_gens, K.lattice())
-
- # Since ``S`` is a subspace, its dual is its orthogonal complement
- # (albeit in the wrong lattice).
- S_perp = Cone(S.dual(), K.lattice())
- P = K.intersection(S_perp)
-
- return (P,S)
-