The first goal of this chapter is to introduce the concept of the intersection body of a star body. Such bodies serve for sections as projection bodies do for projections. The idea of an intersection body is a relatively new one, but the topic has been the focus of intense study in recent years. To facilitate the development, we restrict the discussion to star bodies with continuous radial functions, though some of the theory extends to arbitrary star bodies and indeed to bounded Borel sets (see Notes 8.1 and 8.9).

The radial function of the intersection body of a star body gives the volumes of its intersections with hyperplanes through the origin. The intersection body of a convex body need not be convex, but Busemann's theorem, Theorem 8.1.10, implies that the intersection body of a centered convex body is convex. Examples of intersection bodies of star bodies are centered ellipsoids or any sufficiently smooth centered convex body in En, n ≤ 4 (see Theorem 8.1.17). On the other hand, we see in Theorem 8.1.18 that in four or more dimensions, a cylinder is not the intersection body of a star body.

In the second section, intersection bodies are applied to the Busemann–Petty problem. The analogue of Shephard's problem for projections, this asks whether a centered convex body with central sections of larger volume than another must also have larger volume.