  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, unlike in the spheres, in the Euclidean spaces and the
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 non-constant during the flexion. Hence
these cross-polytopes give counterexamples to the Bellows Conjecture for
spherical polyhedra. Earlier a counterexample to this conjecture was built only
in dimension 3 (Alexandrov, 1997), and 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 among which there are our
counterexamples to the usual Bellows Conjecture. By the way, we obtain several
geometric results on flexible cross-polytopes of the simplest type. In
particular, we write relations on the volumes of their faces of codimensions 1
and 2.
