#### Document Type

Article

#### Publication Date

3-1994

#### Abstract

It is proved that the answer to the Busemann-Petty problem concerning central sections of centrally symmetric convex bodies in *d*-dimensional Euclidean space **E**^{d} is negative for a given *d* if and only if certain centrally symmetric convex bodies exist in **E**^{d} which are not intersection bodies. It is also shown that a cylinder in **E**^{d} is an intersection body if and only if *d* ≤ 4, and that suitably smooth axis-convex bodies of revolution are intersection bodies when *d* ≤ 4. These results show that the Busemann-Petty problem has a negative answer for *d* ≥ 5 and a positive answer for *d* = 3 and *d* = 4 when the body with smaller sections is a body of revolution.

#### Publication Title

Transactions of the American Mathematical Society

#### Volume

342

#### Issue

1

#### First Page

435

#### Last Page

445

#### Required Publisher's Statement

First published in Transactions of the American Mathematical Society in Volume 342, Number 1, March 1994, published by the American Mathematical Society

#### Recommended Citation

Gardner, Richard J., "Intersection Bodies and the Busemann-Petty Problem" (1994). *Mathematics.* Paper 24.

http://cedar.wwu.edu/math_facpubs/24