A new formula for the volume of lattice polyhedra

Abstract

Abstract. Let L, be the lattice consisting of all points x in R N such that nx belongs to the fundamentallattice L 1 of points with integer coordinates. When the vertices of a polyhedron P in R N are restricted to lie in L 1 there is a formula which relates the volume of P to the numbers of points of L1 .... , Lr~ in the interior and on the boundary of P. The aim of this note is to show that the volume of P can be determined only by means of the numbers of points of L1,..., L N lying in the interior of P and cannot be expressed by the numbers of points of L1 .... , L N lying on the boundary of P. The latter numbers in turn can be used to compute the Euler characteristic of the boundary of P.

An ‘odd’ formula for the volume of three-dimensional lattice polyhedra

“…It can be obtained by using formula (3) (2) and (3) applicable to proper lattice polyhedra in R N established by Macdonald [6]. We refer the reader also to [3] and [5]. Note that in [5] it is shown that the volume of a proper lattice polyhedron P in R N can be determined by means only of the numbers of points from L1 ,..., LN lying in the interior of P.…”

“…We refer the reader also to [3] and [5]. Note that in [5] it is shown that the volume of a proper lattice polyhedron P in R N can be determined by means only of the numbers of points from L1 ,..., LN lying in the interior of P.…”