To some extent the theory has grown from attempts to solve specific problems. To illustrate, a few of these are described below.
*Hammer's X-ray problem.* Posed by P. C. Hammer [19] in 1963, this asks how many X-rays are required to permit exact reconstruction of a convex hole in an otherwise homogeneous solid. The question has two parts, depending on whether parallel or point X-rays are used.

Gardner and McMullen [15] gave an answer to the uniqueness aspect of the problem by showing that parallel X-rays of a convex body in *prescribed* finite sets of directions, determine the body uniquely among all convex bodies. In fact, when \(n=2\), they completely classified all such "good" sets of directions and showed that any set of four directions for which the cross-ratio of the four corresponding slopes is a transcendental number is a good set. Later, Gardner and Gritzmann [10] found more "practical" good sets of directions, for example, the four directions parallel to the vectors \((1,0)\), \((0,1)\), \((2,1)\), and \((1,-2)\). The best unrestricted uniqueness theorem for point X-rays was established by Volčič [39]: Every set of four noncollinear points in the plane has the property that X-rays of a planar convex body at these points distinguish it from any other such body. Open problems arise when parallel X-rays of convex bodies in

##### Figure 6. Convex bodies of constant width

*Nakajima's problem.* An old problem dating back to Nakajima's 1926 paper [28] asks whether any convex body in

##### Figure 7. A convex body of constant brightness

*Shephard's problem.* In 1964, Shephard [37] asked the following question. Suppose that \(K\) and \(L\) are origin-symmetric (symmetric with respect to the origin) convex bodies in

\(V_{n-1}(K|u^{\perp})\le V_{n-1}(L|u^{\perp})\),

for each \(u\in S^{n-1}\). Is it true that \(V_n(K)\le V_n(L)\)? In short, must an origin-symmetric body whose shadows have larger areas also have a larger volume? It is easy to see that the answer is affirmative if \(n\le 2\).

##### Figure 8. The ball \(K_1\) has larger volume than the double cone \(K_2\) but projections with smaller areas

The problem was solved independently by Petty [30] and Schneider [35], both of whom showed that the answer is generally negative if \(n\ge 3\), but affirmative for all \(n\) when \(L\) belongs to a special class of bodies called *projection bodies*.

##### Figure 9. Projection bodies of a tetahedron and double cone

*The Busemann-Petty problem.* Suppose that \(K\) and \(L\) are origin-symmetric convex bodies in

\(V_{n-1}(K\cap u^{\perp})\le V_{n-1}(L\cap u^{\perp})\),

for each \(u\in S^{n-1}\). Is it true that \(V_n(K)\le V_n(L)\)? In short, must an origin-symmetric body whose central sections have larger areas also have a larger volume? The question was posed by Busemann and Petty [6] in 1956. It is easy to see that the answer is affirmative if \(n\le 2\).

##### Figure 10. The nonconvex double cone \(L_1\) has larger volume than the ball \(L_2\) but central plane sections with smaller areas

Lutwak [26] proved that it is also affirmative for all \(n\) when \(K\) belongs to a special class of bodies called *intersection bodies*. This insight led, after important contributions by several mathematicians (see [9, Note 8.9]) to a unified solution in all dimensions by Gardner, Koldobsky, and Schlumprecht [14]. The answer is affirmative if \(n\le 4\) and negative when \(n\ge 5\).

##### Figure 11. Intersection bodies of an origin-symmetric cube and cylinder

*The equichordal problem.* For a long time, one of the oldest unsolved problems in planar geometry was the equichordal problem, posed independently by Fujiwara [8] and Blaschke, Rothe, and Weitzenböck [5]. This asks whether there exists a planar body (that is, a compact convex set in

The five problems described above form a small part of the extensive theory surveyed more completely in [9]. Inasmuch as part of geometric tomography concerns projections of convex bodies, the Brunn-Minkowski theory from convex geometry provides many tools, including mixed volumes and a plethora of powerful inequalities such as the isoperimetric inequality and its generalizations. The equally effective dual Brunn-Minkowski theory, initiated by Lutwak in 1975, copes with sections of star bodies in a similar way. There is a rich but still quite mysterious interplay between projections of convex bodies and sections (through the origin) of star bodies. Parallel and point X-rays form another important topic. The theory continues to expand and develop, but most of the 66 open problems stated in [9] remain unanswered.