  We call a continuous path of polygons decreasing if the convex hulls of the
polygons form a decreasing family of sets. For an arbitrary polygon of more
than three vertices, we characterize the polygons contained in it that can be
reached by a decreasing path (attainability problem), and we show that this can
be done by a finite application of "pull-in" moves (bang-bang problem). In the
case of triangles, this problems was investigated by Goodman, Johansen, Ramsey,
and Frydman among others, in connection with the embeddability problem for
non-homogeneous Markov processes.
