The notion of a parallel X-ray of a planar convex body admits several extensions, and the aim of this chapter is to study some of them. One can consider X-rays of convex bodies in higher dimensions. For this Lebesgue measure and integration can be avoided, as in Chapter 1, but when convex bodies are replaced by compact sets, for example, this is no longer possible. In addition, we would also like to discuss higher-dimensional X-rays, in which sections by parallel lines are replaced with sections by parallel planes. In view of this, we begin the chapter with the definitions of the X-ray and k-dimensional X-ray of a bounded Lebesgue measurable set in En, and the corresponding generalizations of the Steiner symmetral.

Although a working knowledge of Lebesgue measure and integration is required for a full understanding of this chapter, much of it can be assimilated with only intuitive ideas of length, area, and volume. Some, but not all, of the background material in the first five sections of Chapter 0 is relevant; this includes a brief introduction to the theory of Lebesgue measure and integration.

Theorem 2.2.5 yields an efficient algorithm for successive determination of a convex polyhedron in E3 by only two X-rays. If S is a subspace, an X-ray of a measurable set parallel to S gives the volumes of the intersections of the set with all translates of S. The s-additive sets are introduced in Definition 2.3.9.