You are not currently logged in.
Access JSTOR through your library or other institution:
If You Use a Screen ReaderThis content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
An Analytic Solution to the Busemann-Petty Problem on Sections of Convex Bodies
R. J. Gardner, A. Koldobsky and T. Schlumprecht
Annals of Mathematics
Second Series, Vol. 149, No. 2 (Mar., 1999), pp. 691-703
Published by: Annals of Mathematics
Stable URL: http://www.jstor.org/stable/120978
Page Count: 13
Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Preview not available
We derive a formula connecting the derivatives of parallel section functions of an origin-symmetric star body in Rn 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 Rn and leads to a unified analytic solution to the Busemann-Petty problem: Suppose that K and L are two origin-symmetric convex bodies in Rn 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 ≤ 4 and the negative answer for n ≥ 5 now follow from the fact that convexity controls the second derivatives, but does not control the derivatives of higher orders.
Annals of Mathematics © 1999 Annals of Mathematics