Dehn Invariant and Scissors Congruence of Flexible Polyhedra

Abstract

We prove that the Dehn invariant of any flexible polyhedron in n-dimensional Euclidean space, where n ≥ 3, is constant during the flexion. For n = 3 and 4 this implies that any flexible polyhedron remains scissors congruent to itself during the flexion. This proves the Strong Bellows Conjecture posed by R. Connelly in 1979. It was believed that this conjecture was disproved by V. Alexandrov and R. Connelly in 2009. However, we find an error in their counterexample. Further, we show that the Dehn invariant of a flexible polyhedron in the n-dimensional sphere or n-dimensional Lobachevsky space, where n ≥ 3, is constant during the flexion whenever this polyhedron satisfies the usual Bellows Conjecture, i.e., whenever its volume is constant during every flexion of it. Using previous results of the first named author, we deduce that the Dehn invariant is constant during the flexion for every bounded flexible polyhedron in odd-dimensional Lobachevsky space and for every flexible polyhedron with sufficiently small edge lengths in any space of constant curvature of dimension at least 3.