On the perfect matching of disjoint compact sets by noncrossing line segments in

Let V be a family of even disjoint line segments in Rn of f-equal width for a direction f∈(Rn)∗(n⩾2), or even disjoint curve segments in Rn of fn-equal width, where fn is the normal direction for bases (n⩾2), or even disjoint twisted triangular prisms in R3 of f3-equal width. We prove that V has a perfect matching by open disjoint line segments in the complementary domain of the union of all the elements of V.