+
+ REFERENCES:
+
+ .. [IusemSeegerOnPairs] Alfredo Iusem and Alberto Seeger.
+ On pairs of vectors achieving the maximal angle of a convex cone.
+ Mathematical Programming, 104(2-3):501-523, 2005,
+ doi:10.1007/s10107-005-0626-z.
+
+ .. [SeegerSossaI] Alberto Seeger and David Sossa.
+ Critical angles between two convex cones I. General theory.
+ TOP, 24(1):44-65, 2016, doi:10.1007/s11750-015-0375-y.
+
+ SETUP::
+
+ sage: from mjo.cone.nonnegative_orthant import nonnegative_orthant
+ sage: from mjo.cone.schur import schur_cone
+
+ EXAMPLES:
+
+ Verify the claim that the maximal angle between any two generators
+ of the Schur cone and the nonnegative quintant is ``3*pi/4``::
+
+ sage: P = schur_cone(5)
+ sage: Q = nonnegative_orthant(5)
+ sage: G = ( g.change_ring(QQbar).normalized() for g in P )
+ sage: H = ( h.change_ring(QQbar).normalized() for h in Q )
+ sage: actual = max(arccos(u.inner_product(v)) for u in G for v in H)
+ sage: expected = 3*pi/4
+ sage: abs(actual - expected).n() < 1e-12
+ True
+
+ TESTS:
+
+ We get the trivial cone when ``n`` is zero::
+
+ sage: schur_cone(0).is_trivial()
+ True
+
+ The Schur cone induces the majorization ordering::
+
+ sage: set_random_seed()
+ sage: def majorized_by(x,y):
+ ....: return (all(sum(x[0:i]) <= sum(y[0:i])
+ ....: for i in range(x.degree()-1))
+ ....: and sum(x) == sum(y))
+ sage: n = ZZ.random_element(10)
+ sage: V = VectorSpace(QQ, n)
+ sage: S = schur_cone(n)
+ sage: majorized_by(V.zero(), S.random_element())
+ True
+ sage: x = V.random_element()
+ sage: y = V.random_element()
+ sage: majorized_by(x,y) == ( (y-x) in S )
+ True
+