On the determination of convex bodies by translates of their projections

It is a well-known fact that a three-dimensional convex body is, up to translations, uniquely determined by the translates of its orthogonal projections onto all planes. Simple examples show that this is no longer true if only 'lateral projections' are permitted, that is orthogonal projections onto all planes that contain a given line. In this article large classes of convex bodies are specified that are essentially determined by translates or homothetic images of their lateral projections. The problem is considered for all dimensions d greater than or equal to 3, and corresponding stability results are proved. Finally, it is investigated to which degree of precision a convex body can be determined by a finite number of translates of its projections. Various corollaries concern characterizations and corresponding stability statements for convex bodies of constant width and spheres.