A convex polygonal region R with k vertices is given on a 2D plane, and there are n convex polygons P

_{i}, with k vertices each, which are pairwise disjoint and completely inside R. These polygons do not sum to R; in other words, there are some gaps inside R that are not part of any of P

_{i}.

Now you want to flip each P

_{i}- place the mirror image P

_{i}' of P

_{i}so that the position of its center of mass remains constant. Each P

_{i}' must still lie inside R and be pairwise disjoint. Rotation of P

_{i}' around the center of mass is freely allowed.

Is it possible to computationally determine whether each polygon can be successfully flipped or not?