  Consider the Riemann sum of a smooth compactly supported function h(x) on a
polyhedron in R^d, sampled at the points of the lattice Z^d/t. We give an
asymptotic expansion when t goes to infinity, writing each coefficient of this
expansion as a sum indexed by the faces f of the polyhedron, where the f-term
is the integral over f of a differential operator applied to the function h(x).
In particular, if a Euclidean scalar product is chosen, we prove that the
differential operator for the face f can be chosen (in a unique way) to involve
only normal derivatives to f. Our formulas are valid for a semi-rational
polyhedron and a real sampling parameter t, if we allow for step-polynomial
coefficients, instead of just constant ones.
