Tangencies between families of disjoint regions in the plane

Abstract

Let C be a family of n convex bodies in the plane, which can be decomposed into k subfamilies of pairwise disjoint sets. It is shown that the number of tangencies between the members of C is at most O ( k n ) , and that this bound cannot be improved. If we only assume that our sets are connected and vertically convex, that is, their intersection with any vertical line is either a segment or the empty set, then the number of tangencies can be superlinear in n, but it cannot exceed O ( n log 2 n ) . Our results imply a new upper bound on the number of regular intersection points on the boundary of ⋃ C .