Tight Bounds for Visibility Matching of f-Equal Width Objects

Abstract

Let s denote a compact convex object in ℝ². The f-width of s is the perpendicular distance between two distinct parallel lines of support of s with direction f. A set of disjoint convex compact objects in ℝ² is of equal f-width if there exists a direction f such that every pair of objects have equal f-width. A visibility matching, for a set of equal f-width objects is a matching using non-crossing lines of site in the visibility graph of the set. In this note we establish tight bounds on the size of a maximal visibility matching for a set of f-equal width objects by showing that $\left\lfloor\frac{2n}{3}\right\rfloor \leq h(n) \leq \frac{2n}{3}$.