New polytope decompositions and Euler–Maclaurin formulas for simple integral polytopes

We give new weighted decompositions for simple polytopes, generalizing previous results of Lawrence–Varchenko and Brianchon–Gram. We start with Witten's non-abelian localization principle in equivariant cohomology for the norm-square of the moment map in the context of toric varieties to obtain a decomposition for Delzant polytopes. Then, by a purely combinatorial argument, we show that this formula holds for any simple polytope. As an application, we study Euler–Maclaurin formulas.