Geometric Permutations of Large Families of Translates

Let F be a finite family of disjoint translates of a compact convex set K in R2, and let l be an ordered line meeting each of the sets. Then l induces in the obvious way a total order on F. It is known that, up to reversals, at most three different orders can be induced on a given F as l varies. It is also known that the families are of six different types, according to the number of orders and their interrelations. In this paper we study these types closely, focusing on their relations to the given set K, and on what happens as |F| →∞.