Properties of the modulus polyhedron

Let in a three-dimensional real space be given three homogeneous real linear forms. Their absolute values give a mapping the space into another space. In the second space we consider the convex hull of images of all integer points of the first space, exept its origin. The convex hull is called the modulus polyhedron. The best integer approximations to the root subspaces of the given forms have images lying in the boundary of the modulus polyhedon. Here we study and prove such properties of the modulus polyhedron, which we use to construct and to justificate our algorithm generalizing the continued fraction.