  We are generalizing to higher dimensions the Bavard-Ghys construction of the
hyperbolic metric on the space of polygons with fixed directions of edges.
  The space of convex d-dimensional polyhedra with fixed directions of facet
normals has a decomposition into type cones that correspond to different
combinatorial types of polyhedra. This decomposition is a subfan of the
secondary fan of a vector configuration and can be analyzed with the help of
Gale diagrams.
  We construct a family of quadratic forms on each of the type cones using the
theory of mixed volumes. The Alexandrov-Fenchel inequalities ensure that these
forms have exactly one positive eigenvalue. This introduces a piecewise
hyperbolic structure on the space of similarity classes of polyhedra with fixed
directions of facet normals. We show that some of the dihedral angles on the
boundary of the resulting cone-manifold are equal to \pi/2.
