Section 5.3 was devoted to point X-rays of planar convex bodies. This chapter takes a wider perspective in two important ways. First, the objects of interest will sometimes be star bodies. The class of star bodies contains the class of convex bodies, and the larger class is not only natural, but, as we shall see in Chapter 8, essential in certain circumstances. Second, we shall consider data of a more general type, the i-chord functions of a star body, where i is a real number. Such a function deals with ith powers of distances from the origin to the boundary of the body, and, when i = 1, reduces to the ordinary X-ray at the origin.

Again, there is a very definite purpose for this generalization. Many of the results in Sections 6.1 and 6.2 will yield, via Theorem 7.2.3 in the next chapter, corresponding information on the determination of bodies by their ith section functions. For example, glance ahead at Corollary 7.2.11, which implies that there are nonspherical convex bodies in E3 with all of their sections by planes through the origin having the same area. The connection is simple; to calculate the area of such a section, one integrates the square of the radial function, so 2-chord functions are implicated.

By allowing i to take other values, problems of the equichordal type can also be treated in a unified manner. The action here centers around the famous equichordal problem.