An analytic solution to the Busemann-Petty problem on sections of convex bodies

Abstract

We derive a formula connecting the derivatives of parallel section functions of an origin-symmetric star body in $\mathbb{R}^n$ 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 $\mathbb{R}^n$ and leads to a unified analytic solution to the Busemann-Petty problem: Suppose that $K$ and $L$ are two origin-symmetric convex bodies in $\mathbb{R}^n$ 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 $n \le 4$ and the negative answer for $n \ge 5$ now follow from the fact that convexity controls the second derivatives, but todes not control the derivatives of higher orders.

DOI
https://doi.org/10.2307/120978

Authors

Richard J. Gardner

Alexander Koldobsky

Thomas Schlumprecht