Convex hull alignment through translation

Given k finite point sets A1, . . . , Ak in R, we are interested in finding one translation for each point set such that the union of the translated point sets is in convex position. We show that if k is part of the input, then it is NP-hard to determine if such translations exist, even when each point set has at most three points. The original motivation of this problem comes from the question of whether a given triangulation T of a point set is the empty shape triangulation with respect to some (strictly convex) shape S. In other words, we want to find a shape S such that the triangles of T are precisely those triangles about which we can circumscribe a homothetic copy of S that does not contain any other vertices of T . This is the Delaunay criterion with respect to S; for the usual Delaunay triangulation, S is the circle.