Integration over facet-simple polytopes

Abstract

AbstractWe present an elementary approach for the computation of integrals of the form $$\int _{\mathcal {P}} f^{(n)}(\textbf{s} \cdot \textbf{x})\,\textbf{dx}$$ ∫ P f ( n ) ( s · x ) dx over polytopes $$\mathcal {P}$$ P , where $$f: \mathbb {C}\rightarrow \mathbb {C}$$ f : C → C is analytic. The proof is based on an independence theorem on exponential functions over the field of rational functions and needs only simple facts from the theory of polyhedra. In particular we present an explicit formula for generalized facet-simple polytopes. Here a convex polytope is called facet-simple if each of its facets is simple and a set of points is called a generalized facet-simple polytope if it is a finite union of n-dimensional facet-simple convex polytopes such that any two distinct members are either disjoint or intersect in a common facet.