  This is the first of two companion papers. The joint aim is to study a
generalization to higher dimension of the point vortex systems familiar in 2-D.
In this paper we classify the momentum polytopes for the action of the Lie
group SU(3) on products of copies of complex projective 4-space. For 2 copies,
the momentum polytope is simply a line segment, which can sit in the positive
Weyl chamber in a small number of ways. For a product of 3 copies there are 8
different types of generic momentum polytope, and numerous transition
polytopes, all of which are classified here. The type of polytope depends on
the weights of the symplectic form on each copy of projective space. In the
second paper we use techniques of symplectic reduction to study the possible
dynamics of interacting generalized point vortices. The results can be applied
to determine the inequalities satisfied by the eigenvalues of the sum of up to
three 3x3 Hermitian matrices with double eigenvalues.
