Bounded polytopes without ''excess rigidity'' in Lobachevskij spaces (Q1177556)

The construction of discrete motion groups generated by reflections in Lobachevskij space \(\Lambda^ n\) reduces to the construction of their fundamental regions --- Coxeter polytopes. These are convex polytopes whose dihedral angles have measures \(\pi/n\), \(n\) an integer at least 2. Such polytopes, when bounded, are simple in that each face of co- dimension \(m\), lies in exactly \(m\) facets. In what follows all polytopes are assumed to have all the properties just described. Such polytopes in \(\Lambda^ n\), \(n>2\), are determined uniquely, up to a motion, from their dihedral angles. However for dimensions \(n>3\), the number of degrees of freedom for such a polytope can be less than the number of its dihedral angles and, in such cases, the polytope is said to have redundant stiffness. If \(a_ k\) denotes the number of \(k\)-dimensional faces in the polytope \(P\), then \(\Sigma(P)=a_{n-2}-n a_{n- 1}+n(n+1)/2\) measures the redundant stiffness of \(P\); \(\Sigma(P)\geq 0\) if \(n\geq 3\); when \(\Sigma(P)=0\), \(P\) is said to be without redundant stiffness. \(\Sigma(P)\) does not change under the cutting off of any one vertex by a hyperplane; also a simplex is a polytope with \(\Sigma(P)=0\). È. B. Vinberg conjectured that for \(n\geq 4\), each polytope without redundant stiffness can be reduced to a simplex through a finite sequential cutting off of vertices by hyperplanes. The present paper proves this conjecture under the added assumption that there is a polytope without redundant stiffness which is combinatorially equivalent to the polytope under consideration.