  We give new bounds on the Erdos-Szekeres theorems for convex bodies of
Bisztriczky and Fejes Toth and of Pach and Toth. We derive them from a
combinatorial characterization of convex position of a family of planar convex
bodies. This characterization confirms that the concept of Order Type for
points can be extended to noncrossing families of convex bodies in a
geometrically meaningful way.
