Theory

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 R\(^n\), taken in certain 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 R\(^n\), \(n\ge 3\), are taken in directions in general position, while for point X-rays even the planar case is not fully understood.

Convex Bodies of Constant Width 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 R\(^3\) of constant width and constant brightness (that is, whose width and brightness functions are constant) must be a ball. Over the years this question has been the motivation of much work. For example, Goodey, Schneider, and Weil [18] demonstrate that most (in the sense of Baire category) convex bodies in R\(^n\) are determined, up to translation and reflection in the origin, by their width and brightness functions. The problem was finally solved by Howard [22], who gave an affirmative answer. The same question can be asked for convex bodies in R\(^n\), \(n\ge 4\), and in this case it is still unsolved (see [23] for further results).

Convex Bodies of Constant Brightness 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 R\(^n\), such that

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

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 \(n\le 2\).

Shephard's Problem 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.

Projection Bodies of a Tetrahedron and a Double Cone 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 R\(^n\), such that

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

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 \(n\le 2\).

Nonconvex Double Cone and Ball 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 \(n\ge 5\).

Intersection Bodies of Cube and Cylinder

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 R\(^2\) equal to the closure of its interior) that contains two different equichordal points. Here, an equichordal point \(p\) in the interior of a body \(E\) is one such that \(E\) is star-shaped with respect to \(p\) and each chord of \(E\) containing \(p\) has the same length. For example, the center of a disk is an equichordal point of it. The book [25, Problem 2] of Klee and Wagon contains an excellent report on the history of the problem up to 1991. Finally, in 1997, Rychlik [34] solved the equichordal problem by showing that no such body exists. The long argument uses invariant manifold theory and some fairly heavy machinery from complex function theory.

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.