  We give an alternative proof of the main result of the paper
http://arxiv.org/abs/math/0112104, the proof relies on Brion's theorem about
convex polyhedra. The result itself can be viewed as a formula for the
character of the Feigin-Stoyanovsky subspace of an integrable irreducible
representation of the affine Lie algebra
$\widehat{\mathfrak{sl}_n}(\mathbb{C})$. Our approach is to assign integer
points of a certain polytope to the vectors comprising a monomial basis of the
subspace and then compute the character via (a variation of) Brion's theorem.
