  We give a local Euler-Maclaurin formula for rational convex polytopes in a
rational euclidean space . For every affine rational polyhedral cone C in a
rational euclidean space W, we construct a differential operator of infinite
order D(C) on W with constant rational coefficients, which is unchanged when C
is translated by an integral vector. Then for every convex rational polytope P
in a rational euclidean space V and every polynomial function f (x) on V, the
sum of the values of f(x) at the integral points of P is equal to the sum, for
all faces F of P, of the integral over F of the function D(N(F)).f, where we
denote by N(F) the normal cone to P along F.
