Embedded flexible spherical cross-polytopes with nonconstant volumes

We construct examples of embedded flexible cross-polytopes in the spheres of all dimensions. These examples are interesting from two points of view. First, in dimensions $4$ and higher, they are the first examples of embedded flexible polyhedra. Notice that, in contrast to the spheres, in the Euclidean and Lobachevsky spaces of dimensions $4$ and higher still no example of an embedded flexible polyhedron is known. Second, we show that the volumes of the constructed flexible cross-polytopes are nonconstant during the flexion. Hence these cross-polytopes give counterexamples to the Bellows Conjecture for spherical polyhedra. Earlier a counterexample to this conjecture was constructed only in dimension $3$ (V. A. Alexandrov, 1997), and it was not embedded. For flexible polyhedra in spheres we suggest a weakening of the Bellows Conjecture, which we call the Modified Bellows Conjecture. We show that this conjecture holds for all flexible cross-polytopes of the simplest type, which includes our counterexamples to the ordinary Bellows Conjecture. Simultaneously, we obtain several geometric results on flexible cross-polytopes of the simplest type. In particular, we write down relations for the volumes of their faces of codimensions $1$ and $2$.