Document Type

Article

Publication Date

3-1999

Keywords

Convex body, star body, Busemann-Petty problem, intersection body, Fourier transform

Abstract

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.

Publication Title

Annals of Mathematics

Volume

149

Issue

2

First Page

691

Last Page

703

DOI

http://dx.doi.org/10.2307/120978

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

Subjects - Topical (LCSH)

Convex bodies; Convex geometry; Intersection theory (Mathematics); Variable stars; Fourier transformations; Radon transforms

Genre/Form

articles

Type

Text

Rights

Copying of this document in whole or in part is allowable only for scholarly purposes. It is understood, however, that any copying or publication of this document for commercial purposes, or for financial gain, shall not be allowed without the author’s written permission.

Language

English

Format

application/pdf

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Included in

Mathematics Commons

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