An analytic solution to the BusemannPetty problem on sections of convex bodies
Abstract
We derive a formula connecting the derivatives of parallel section functions of an originsymmetric star body in R^n with the Fourier transform of powers of the radial function of the body. A parallel section function (or (n1)dimensional Xray) gives the ((n1)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 BusemannPetty problem: Suppose that K and L are two originsymmetric convex bodies in R^n such that the ((n1)dimensional) volume of each central hyperplane section of K is smaller than the volume of the corresponding section of L; is the (ndimensional) volume of K smaller than the volume of L? In conjunction with earlier established connections between the BusemannPetty problem, intersection bodies, and positive definite distributions, our formula shows that the answer to the problem depends on the behavior of the (n2)nd derivative of the parallel section functions. The affirmative answer to the BusemannPetty 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 February 1999
 arXiv:
 arXiv:math/9903200
 Bibcode:
 1999math......3200G
 Keywords:

 Mathematics  Metric Geometry
 EPrint:
 13 pages, published version