Families of Convex Sets not Representable by Points

Let (A,B,C) be a triple of disjoint closed convex sets in the plane such that each of them contributes at least one point to the boundary partial derivative of the convex hull of their union. If there are three points a is an element of A,b is an element of B,c is an element of C that belong to partial derivative and follow each other in clockwise (counterclockwise) order, we say that the orientation of the triple (A,B,C) is clockwise (counterclockwise). We construct families of disjoint closed convex sets {C-1, ..., C-n} in the plane whose every triple has a unique orientation, but there are no points p(1), ..., p(n) in general position in the plane whose triples have the same orientations. In other words, these families cannot be represented by a point set of the same order type. This answers a question of A. Hubard and L. Montejano. We also show the size of the largest subfamily representable by points, which can be found in any family of n disjoint closed convex sets in general position in the plane, is O(n(log8/log9)). Some related Ramsey-type geometric problems are also discussed.