  The weights of finite-dimensional representations of simple Lie algebras are
naturally associated with Weyl polytopes. Representation characters decompose
into multiplicity-free sums over the weights in Weyl polytopes. The Brion
formula for these Weyl polytope sums is remarkably similar to the Weyl
character formula. Moreover, the same Lie characters are also expressible as
Demazure character formulas. This motivates a search for new expressions for
Weyl polytope sums, and we prove such a formula involving Demazure operators.
It applies to the Weyl polytope sums of the simple Lie algebras $A_n$, for all
dominant integrable highest weights and all ranks $n$.
