A new formula for the volume of lattice polyhedra

LetL_n be the lattice consisting of all pointsx inR^N such thatnx belongs to the fundamental latticeL₁ of points with integer coordinates. When the vertices of a polyhedronP inR^N are restricted to lie inL₁ there is a formula which relates the volume ofP to the numbers of points ofL₁,...,L_N in the interior and on the boundary ofP. The aim of this note is to show that the volume ofP can be determined only by means of the numbers of points ofL₁,...,L_N lying in the interior ofP and cannot be expressed by the numbers of points ofL₁,...,L_N lying on the boundary ofP. The latter numbers in turn can be used to compute to comopute the Euler characteristic of the boundary ofP.