A Klein polyhedron is defined as the convex hull of nonzero lattice points inside an orthant of ${ℝ}^n$. It generalizes the concept of continued fraction. In this paper facets and edge stars of vertices of a Klein polyhedron are considered as multidimensional analogs of partial quotients and quantitative characteristics of these “partial quotients”, so called determinants, are defined. It is proved that the facets of all the $2^n$ Klein polyhedra generated by a lattice $\Lambda$ have uniformly bounded determinants...