Intersection bodies and the Busemann-Petty problem

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