  The characters of simple Lie algebras are naturally decomposed into lattice
polytope sums. The Brion formula for those polytope sums is remarkably similar
to the Weyl character formula. Here we start to investigate if other character
formulas have analogs for lattice polytope sums, by focusing on the Demazure
character formulas. Using Demazure operators, we write expressions for the
lattice sums of the weight polytopes of rank-2 simple Lie algebras, and the
rank-3 algebra $A_3$.
