The Lower Dimensional Busemann-petty Problem for Bodies with the Generalized Axial Symmetry

The lower dimensional Busemann-Petty problem asks, whether n-dimensional origin-symmetric convex bodies, having smaller i-dimensional sections, necessarily have smaller volumes. For i = 1, the affirmative answer is obvious. For i > 3, the answer is negative. For i = 2 and i = 3, the problem is still open, except when the body with smaller sections is a body of revolution. In this case the answer is affirmative. The paper contains a complete solution to the problem in the more general situation, when the body with smaller sections is invariant under orthogonal transformations preserving coordinate subspaces R and R of R for arbitrary fixed 0 < l < n.