Ghosts are Scarce

Abstract

In recent years P. C. Hammer's problem [8] of determining a convex body from its 'X-ray pictures' was investigated by Gardner and McMullen [4], Gardner [3], Falconer [2] and Volcic [15]. An earlier result is due to Giering [5]. An X-ray picture of a convex body in a direction may be identified with its Steiner symmetral in that direction. Some of these papers consider X-ray pictures taken from points not on the line at infinity, but here we are not concerned with that situation. Gardner and McMullen proved that there exist four directions such that the corresponding X-ray pictures distinguish between all convex bodies, and that no three directions can do this. Giering proved that, given a plane convex body K, there exist three directions depending on K, such that the corresponding X-ray pictures distinguish K from any other convex body. He has also shown that two directions are in general not enough. Convex bodies with the same X-ray pictures as a given one were called 'ghosts' in [14], in analogy with the ghost densities from computerized tomography [12]. It should be remembered that in the fundamental case of parallel rays from two orthogonal directions, besides a few triangular or quadrangular examples by Giering [6] and a rather obvious construction which basically interchanges two diagonally opposite, symmetrical pieces with two other diagonally opposite congruent pieces (diagonals of a rectangle), no deeper insight into the soul of a ghost of a convex body has been won. We are—as a consequence—far away from being able to characterize convex are not ghosts! (Note the equivalence between having and being a ghost!) Thus we In this situation, the question about the generic behaviour of convex bodies with regard to their ghosts appears interesting, but looks at a first glance, in view of the lack of knowledge described above, rather hopeless. However, in this paper we establish the validity of the (more comfortable?) assertion that most convex bodies are not ghosts! (Note the equivalence between having and being a ghost!) Thus we confirm a conjecture of the first author, motivated by the symmetries described above and also present in his examples from [14]. It is clear that the orthogonality of the two considered directions is unessential, because of the afrlne character of our problem. When we state it we do so just to fix the ideas. As a main open problem there remains the characterization of those convex bodies which are uniquely determined by two X-ray pictures. The analogous problem for measurable sets has been solved by Lorentz [11]. The space # of all convex curves in 1R, like the space & of all convex bodies in U, equipped with the Hausdorff distance S is a Baire space. 'Most ' means 'all, except those in a set of first category'. For a survey on properties of most convex bodies, see [16].