+
+
+def has_rearrangement_property(v, p):
+ r"""
+ Test if the vector ``v`` has the "rearrangement property."
+
+ The rearrangement cone of order ``p`` in `n` dimensions has its
+ members vectors of length `n`. The "rearrangement property,"
+ satisfied by its elements, is to have its smallest ``p`` components
+ sum to a nonnegative number.
+
+ We believe that we have a description of the extreme vectors of the
+ rearrangement cone: see ``rearrangement_cone()``. This function is
+ used to test that conic combinations of those extreme vectors are in
+ fact elements of the rearrangement cone. We can't test all conic
+ combinations, obviously, but we can test a random one.
+
+ To become more sure of the result, generate a bunch of vectors with
+ ``random_element()`` and test them with this function.
+
+ INPUT:
+
+ - ``v`` -- An element of a cone suspected of being the rearrangement
+ cone of order ``p``.
+
+ - ``p`` -- The suspected order of the rearrangement cone.
+
+ OUTPUT:
+
+ If ``v`` has the rearrangement property (that is, if its smallest ``p``
+ components sum to a nonnegative number), ``True`` is returned. Otherwise
+ ``False`` is returned.
+
+ SETUP::
+
+ sage: from mjo.cone.rearrangement import (has_rearrangement_property,
+ ....: rearrangement_cone)
+
+ EXAMPLES:
+
+ Every element of a rearrangement cone should have the property::
+
+ sage: set_random_seed()
+ sage: all( has_rearrangement_property(
+ ....: rearrangement_cone(p,n).random_element(),
+ ....: p
+ ....: )
+ ....: for n in range(2, 10)
+ ....: for p in range(1, n-1)
+ ....: )
+ True
+
+ """
+ components = sorted(v)[0:p]
+ return sum(components) >= 0