On the volume of lattice manifolds

The volume of a general lattice polyhedron P in ℝN can be determined in terms of numbers of lattice points from N − 1 different lattices in P Ehrhart gave a formula for the volume of “polyèdre entier” in even-dimensional spaces involving only N/2 lattices. The aim of this note is to comment on Ehrhart's formula and provide a similar volume formula applicable to lattice polyhedra that are N-dimensional manifolds in ℝN.

The volume of a lattice polyhedron

Let L be the lattice of all points with integer coordinates in the real affine plane R2 (with respect to some fixed coordinate system). Let X be a finite rectilinear simplicial complex in R2 whose 0-simplexes are points of L. Suppose X is pure and the frontier Ẋ of X is a Jordan curve; then there is a well-known formula for the area of X in terms of the number of points of L which lie in X and Ẋ respectively, namelywhere L(X) (resp. L(Ẋ)) is the number of points of L which lie in X (resp. Ẋ), and μ(X) is the area of X, normalized so that a fundamental parallelogram of L has unit area.