#### Document Type

Article

#### Publication Date

3-1999

#### Abstract

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

*and leads to a unified analytic solution to the Busemann-Petty problem: Suppose that*

^{n}*K*and

*L*are two origin-symmetric convex bodies in R

*such that the ((*

^{n}*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.

#### Publication Title

Annals of Mathematics

#### Volume

149

#### Issue

2

#### First Page

691

#### Last Page

703

#### Required Publisher's Statement

Published by: Annals of Mathematics

Princeton University, Department of Mathematics

Article DOI: 10.2307/120978

Stable URL: http://www.jstor.org/stable/120978

#### Recommended Citation

Gardner, Richard J.; Koldobsky, A.; and Schlumprecht, T., "An Analytic Solution to the Busemann-Petty Problem on Sections of Convex Bodies" (1999). *Mathematics.* Paper 17.

http://cedar.wwu.edu/math_facpubs/17