On an estimate of the size of the maximum matching for a family of disjoint compact convex sets in the plane

Let C be any family of 2n disjoint compact convex sets in the plane. If C has f-equal width for some direction f, then ⌊2n/3⌋ pairs of elements in C can always be matched by disjoint line segments and more than ⌊4n/5⌋ pairs cannot occasionally be matched. Furthermore, if C is a family of translates of the set satisfying a certain constraint, then ⌊4n/5⌋ pairs of elements can always be matched by disjoint line segments.