Convex hull alignment through translation

Givenkfinite point setsA 1 ,...,A k inR 2 , 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 ifkis 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 triangulationTof a point set is theempty shape triangulationwith respect to some (strictly convex) shapeS. In other words, we want to find a shapeSsuch that the triangles ofTare precisely those triangles about which we can circumscribe a homothetic copy ofSthat does not contain any other vertices ofT. This is the Delaunay criterion with respect toS; for the usual Delaunay triangulation, Sis the circle.