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. © 2007 Elsevier Inc. All rights reserved.
Agapito, J., & Godinho, L. (2007). New polytope decompositions and Euler-Maclaurin formulas for simple integral polytopes. Advances in Mathematics, 214(1), 379–416. https://doi.org/10.1016/j.aim.2007.02.008