Intersection bodies, positive definite distributions, and the Busemann-Petty problem

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 ℓⁿ_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.