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 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 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 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 ≤ 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.

The Brunn-Minkowski inequality

“…Keith Ball started these rolling with his elegant application of the Brascamp-Lieb inequality (59) to the volume of central sections of the cube and to a reverse isoperimetric inequality (67). In the same camp as the latter is Milman's reverse Brunn-Minkowski inequality (68), which features prominently in the local theory of Banach spaces. The whole story extends far beyond Figure 1 and the previous paragraph.…”

“…The answer is no in general in five or more dimensions, but yes in less than five dimensions. See [64], [65], [68], [152], and [154].…”

Gaussian Brunn-Minkowski inequalities

“…Around 1975, Lutwak [27] observed that when the Minkowski sum of two sets is replaced by an operation he called radial sum, in which only sums of parallel vectors are taken into account, a theory arises that is ideal for treating metrical problems about sets star-shaped with respect to the origin, and their intersections with subspaces. This newer theory, now called the dual Brunn-Minkowski theory, has attracted much attention and counts among its successes the solution of the 1956 Busemann-Petty problem on volumes of central sections of o-symmetric convex bodies; see [15], [16], [17], [26], [30], and [31].…”

Konvexgeometrie

“…Our main tool is the Fourier analytic inversion formula from [GKS2]. It allows to obtain the results for zonoids together with the results about the intersection bodies.…”