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We prove that the Dehn invariant of any flexible polyhedron in Euclideanspace of dimension greater than or equal to 3 is constant during the flexion.In dimensions 3 and 4 this implies that any flexible polyhedron remainsscissors congruent to itself during the flexion. This proves the Strong BellowsConjecture posed by Connelly in 1979. It was believed that this conjecture wasdisproved by Alexandrov and Connelly in 2009. However, we find an error intheir counterexample. Further, we show that the Dehn invariant of a flexiblepolyhedron in either sphere or Lobachevsky space of dimension greater than orequal to 3 is constant during the flexion if and only if this polyhedronsatisfies the usual Bellows Conjecture, i.e., its volume is constant duringevery flexion of it. Using previous results due to the first listed author, wededuce that the Dehn invariant is constant during the flexion for every boundedflexible polyhedron in odd-dimensional Lobachevsky space and for every flexiblepolyhedron with sufficiently small edge lengths in any space of constantcurvature of dimension greater than or equal to 3.

Dehn invariant of flexible polyhedra