  We derive a formula connecting the derivatives of parallel section functions
of an origin-symmetric star body in R^n with the Fourier transform of powers of
the radial function of the body. A parallel section function (or
(n-1)-dimensional X-ray) gives the ((n-1)-dimensional) volumes of all
hyperplane sections of the body orthogonal to a given direction. This formula
provides a new characterization of intersection bodies in R^n and leads to a
unified analytic solution to the Busemann-Petty problem: Suppose that K and L
are two origin-symmetric convex bodies in R^n such that the ((n-1)-dimensional)
volume of each central hyperplane section of K is smaller than the volume of
the corresponding section of L; is the (n-dimensional) volume of K smaller than
the volume of L? In conjunction with earlier established connections between
the Busemann-Petty problem, intersection bodies, and positive definite
distributions, our formula shows that the answer to the problem depends on the
behavior of the (n-2)-nd derivative of the parallel section functions. The
affirmative answer to the Busemann-Petty problem for n\le 4 and the negative
answer for n\ge 5 now follow from the fact that convexity controls the second
derivatives, but does not control the derivatives of higher orders.
