Determination of Q-convex sets by X-rays

In this paper, the problem of the determination of lattice sets from X-rays is studied. We define the class of Q-convex sets along a set D of directions which generalizes classical lattice convexity and we prove that for any D, the X-rays along D determine all the convex sets if and only if it determines all the Q-convex sets along D. As a consequence, any algorithm which reconstructs Q-convex sets from X-rays can be used to reconstruct convex lattice sets from X-rays along directions which provide uniqueness. This gives a constructive answer to the discrete version of Hammer's X-ray problem.