  We introduce a characteristic vector with respect to a regular triangulation
of the momentum polytope to compute the Hurwitz polytope of a given smooth
toric variety. As a result, we prove that the convex hull of such vectors of
all regular triangulations is included in the Hurwitz polytope of a smooth
toric surface. In addition, we discuss the relations of such vectors to
K-stability of pairs by Paul and toric K-stability by Donaldson.
