On the statistical properties of Klein polyhedra in three-dimensional lattices (Q2848650)

Let \(L\) be a complete three-dimensional lattice in \(\mathbb{R}^3\) and let \(O\) be one of the orthants in \(\mathbb{R}^3\). The boundary of the convex hull of all lattice points of \(L\) except for the origin is called a Klein polyhedron of the lattice \(L\). In fact, the Klein polyhedron is the set with the polyhedral structure. In this article the author obtains the formulae for the average value of the number of faces of a fixed type and of vertices of Klein polyhedra of three-dimensional lattice with a given determinant.