Spherical complexes and radial projections of polytopes

Using the techniques of Gale diagrams a simple criterion is given for determining when a given spherical complex onS ⁿ⁻¹ ⊂E ⁿ is the radial projection, from the centre ofS ⁿ⁻¹, of a convex polytope. Previously a criterion was known only for the casen=2.