ON THE RECONSTRUCTION OF BODIES FROM THEIR PROJECTIONS OR SECTIONS

We consider two long-standing open problems of geometric tomography:Problem 1: Let k,d be two fixed integers, 2=k=d-1. Assume that K and L are convex bodies in Euclidean space E^d such that their projections K|H and L|H are congruent for each subspace H, dim¿(H) = k. Is L a translate of K or -K?Problem 2: Let k,d be two fixed integers, 2=k=d-1. Assume that K and L are star bodies in Euclidean space E^d such that their sections KnH and LnH are congruent for each subspace H, dim¿(H) = k. Does it imply that L=K or L=-K?We give the affirmative answers to Problems 1 and 2 in the class of convex polytopes. We also prove an affirmative result related to Problem 1: two classical hedgehogs (a class of possibly singular, self-intersecting and non-convex surfaces) coincide up to a translation and a reflection in the origin, provided that their projections onto any 2-dim subspace are directly congruent and have no direct rigid motion symmetries. This is a consequence of a more general analytic statement about the solutions of a functional equation.