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 ∂ of the convex hull of their union. If there are three points a ∈ A, b ∈ B, c ∈ C that belong to ∂ 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 {C1, . . . , Cn} in the plane whose every triple has a unique orientation, but there are no points p1, . . . , pn 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 8/ log ). Some related Ramsey-type geometric problems are also discussed.