A criterion for the integral equivalence of two generalized convex integer polyhedra

We introduce the notion of integral equivalence and formulate a criterion for the equivalence of two polyhedra having certain special properties. The category of polyhedra under consideration includes Klein polyhedra, which are the convex hulls of nonzero points of the lattice ?³ that belong to some 3-dimensional simplicial cone with vertex at the origin, and therefore the criterion enables one to improve some results related to Klein polyhedra. In particular, we suggest a simplified formulation of a geometric analog of Lagrange’s theorem on continued fractions in the three-dimensional case.