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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.

We consider several generalizations of the concept of an intersection body and show their connections with the Fourier transform and embeddings in L p -spaces. These connections lead to generalizations of the recent solution to the Busemann-Petty problem on sections of convex bodies.

Abstract. We consider the following problem. Does there exist an absolute constant C so that for every n ∈ N, every integer 1 ≤ k < n, every origin-symmetric convex body L in R n , and every measure µ with non-negative even continuous density in R n ,where Gr n−k is the Grassmanian of (n − k)-dimensional subspaces of R n , and |L| stands for volume? This question is an extension to arbitrary measures (in place of volume) and to sections of arbitrary codimension k of the slicing problem of Bourgain, a major open problem in convex geometry.It was proved in [K4, K5] that (1) holds for arbitrary originsymmetric convex bodies, all k and all µ with C ≤ O( √ n). In this article, we prove inequality (1) with an absolute constant C for unconditional convex bodies and for duals of bodies with bounded volume ratio. We also prove that for every λ ∈ (0, 1) there exists a constant C = C(λ) so that inequality (1) holds for every n ∈ N, every origin-symmetric convex body L in R n , every measure µ with continuous density and the codimension of sections k ≥ λn. The proofs are based on a stability result for generalized intersection bodies and on estimates of the outer volume ratio distance from an arbitrary convex body to the classes of generalized intersection bodies. In the last section, we show that for some measures the behavior of minimal sections may be very different from the case of volume.

The 1956 Busemann-Petty problem asks whether symmetric convex bodies with larger central hyperplane sections also have greater volume. In 1988, Lutwak introduced the concept of an intersection body which is closely related to the Busemann-Petty problem. We prove that an origin-symmetric star body K in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] is an intersection body if and only if ║ x ║ -1 K is a positive definite distribution on [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /], where ║ x ║ K = min{ a > 0 : x ∈ aK }. We use this result to show that for every dimension n there exist polytopes in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] which are intersection bodies (for example, the cross-polytope), the unit ball of every subspace of L p , 0 < p ≤ 2 is an intersection body, the unit ball of the space ℓ n q , 2 < q < ∞, is not an intersection body if n ≥ 5. Using Lutwak's connection with the Busemann-Petty problem, we present new counterexamples to the problem for n ≥ 5, and confirm the conjecture of Meyer that the answer to the problem is affirmative if the smaller body is a polar projection body.