An ''odd'' formula for the volume of three-dimensional lattice polyhedra
Let \(L_1\) be the fundamental lattice with integer coordinates in \(\mathbb{R}^3\), and let \(L_n\) be the lattice consisting of all points \(x\) in \(\mathbb{R}^3\) such that \(nx\) belongs to \(L_1\). For proper lattice polyhedra, i.e. polyhedra which have their vertices in \(L_1\) and which are equal to the closure of their interior, \textit{J. E. Reeve} [Proc. Lond. Math. Soc., II. Ser. 7, 378-395 (1957; Zbl 0080.26701)] established a formula which expresses the volume \(V(P)\) in terms of \(B_n= |L_n\cap \partial P|\), \(I_n= |L_n \cap \text{int} P|\) and the Euler characteristics \(\chi(P)\) and \(\chi (\partial P)\). In the simplest case of the lattices \(L_1\) and \(L_2\) this formula involves 27 points in each unit cube. The author establishes an analogous formula which involves only 9 points in each unit cube. This formula uses only the number of points lying in the interior and in the relative interior of the faces of proper lattice polyhedra. Moreover it does not make use of the Euler characteristic. Its proof is a modification of the proof of Reeve's formula. The proof is slightly shorter as it is sufficient to count interior points and points in the relative interiors of the faces of \(P\).
scientific article; zbMATH DE number 912996
scientific article