Lattice points in simple polytopes

Abstract

in terms of fP(h) q(x)dx where the polytope P(h) is obtained from P by independent parallel motions of all facets. This extends to simple lattice polytopes the EulerMaclaurin summation formula of Khovanskii and Pukhlikov [8] (valid for lattice polytopes such that the primitive vectors on edges through each vertex of P form a basis of the lattice). As a corollary, we recover results of Pommersheim [9] and Kantor-Khovanskii [6] on the coefficients of the Ehrhart polynomial of P. Our proof is elementary. In a subsequent article, we will show how to adapt it to compute the equivariant Todd class of any complete toric variety with quotient singularities. The Euler-Maclaurin summation formula for simple lattice polytopes has been obtained independently by Ginzburg-Guillemin-Karshon [4]. They used the dictionary between convex polytopes and projective toric varieties with an ample divisor class, in combination with the Riemann-Roch-Kawasaki formula ([1], [7]) for complex manifolds with quotient singularities. A counting formula for lattice points in lattice simplices has been announced by Cappell and Shaneson [2], as a consequence of their computation of the Todd class of toric varieties with quotient singularities.