5. Open Problems 6. References 4. Case 2: P and Q Are Arbitrary Crossing Polygons

Finding the minimum vertex distance between two dis-joint convex polygons in linear time. [6] Supowit, K. J.: The relative neighborhood graph, with an application to minimum spanning trees. A note on the all-nearest-neighbor problem for convex polygons. [10] O'Rourke, J., et al.: A new linear algorithm for intersecting convex polygons.Since P nw and Q R are two linearly separable convex polygons d min (P nw , Q R) can be solved with the techniques of [4] and [5]. Thus we turn our attention to d min (P nw , Q L). We can decompose this problem into two subproblems by splitting Q L into two convex polygons Q Lout and Q L-in , whose vertices lie outside P nw and inside P nw , respectively. We can determine all the sub-chains of Q L that lie inside and outside P nw , and thus Q Lout and Q L-in , by applying the simple linear algorithm of O'Rourke et al. [10] to intersect the two convex polygons P nw and Q L. We are left to solve for Now d min (P nw , Q Lout) is taken care of by theorem 2.1. Finally, since Q L-in lies completely inside P nw , d min (P nw , Q L-in) is nothing but case 1 revisited. Therefore case 2 can also be solved in O(m+n) time. It is possible to determine in O(log (m+n)) time whether the interiors of P and Q intersect or not [11]. If the interiors intersect it is more difficult to determine whether one polygon is entirely inside another and, in fact, Chazelle [9] has proved an Ω(m+n) lower bound for this problem. However, using the linear intersection algorithm of O'Rourke et al. [10] we can solve this problem in O(m+n) time by merely checking to see if all the vertices of P ∩ Q belong to only one of there polygons. We therefore have the following result. Theorem 4.1: The minimum vertex-distance between two convex polygons P and Q of m and n vertices, respectively, can be computed in O(m+n) time. Several interesting problems remain. One pertains to three dimensions. Given two convex polyhedra in three dimensions is it possible to compute the minimum vertex distance in o(mn) time. Another open question concerns the planar all-nearest-distance-between-sets problem. Here, given two convex polygons P and Q we want to find, in O(m+n) time, for each vertex …