Approximation of convex bodies from chord functions

Abstract

In this paper we describe a method for computing derivatives of the polar representations of the boundary of a convex body, from generalized chord functions at two points. INTRODUCTIONThe subject of this paper is related to the large range of questions arising from Hammer's X-ray problems [5].Let us consider the problem of reconstructing a planar convex body K, when, for an integer n, generalized n-chord functions from two distinct points P1 and P2, not lying on its boundary, are assigned. In the context of Hammer's X-ray problems, such points may represent two X-ray sources. Results about uniqueness of solutions, in the cases of 'sources' both interior or both exterior to K, can be found in some papers of Falconer ([1], [2]), Gardner ([4]), and Vol~i~ ([7]). The n-chord functions have been considered by Falconer ([1], [2]) and, in a generalized form, by Gardner ([4]); we use Gardner's definition, adapted to the case of angles between half-lines. We suppose that the straight line joining the 'sources' intersects the boundary of K at two distinct points. Methods for computing these intersection points are produced by Falconer ([1], [2]) for generalized n-chord functions, with n ~> 1. These techniques can be adapted also to the case n < 1. We observe that only for n = 1 does a global study of the given n-chord functions appear to be essential, otherwise these points can be obtained by solving a system of algebraic equations.We continue, in some sense, Falconer's program: we provide methods to evaluate the jth derivatives of the polar representations of the boundary, at the points Z1, Z z where OK intersects the line P1P2. Our main result is the following: if the boundary is sufficiently smooth and the sources are both interior or both exterior to K, then thejth derivatives are uniquely determined from two given n-chord functions, except forj = 1 -n (ifn ~< 0). We also study the case in which one and only one 'source' is interior to K and we obtain similar results, except when the 'sources' and the intersection points ZI, Z2 form a harmonic group.The described process is, at least theoretically, constructive; it provides an alternative proof of a theorem of Falconer about uniqueness ([1, Th. 1]), if the boundary is supposed analytic and n > 0.Furthermore, these results show that some assumptions of a theorem of Gardner ([4, Th. 4]) are redundant.