Cutting glass

J. Urrutia asked the following question. Given a family of pairwise disjoint compact convex sets on a sheet of glass, is it true that one can always separate from one another a constant fraction of them using edge-to-edge straight-line cuts? We answer this question in the negative, and establish some lower and upper bounds for the number of separable sets. In particular, we show that any family F of n pairwise disjoint convex polygons has at least n 1/3 separable members, and a subfamily with this property can be constructed in O(N + n log n) time, where N denotes the total number of sides of F. We also consider the special cases when the family consists of intervals, axis-parallel rectangles, 'fat' sets, or 'fat' sets with bounded size. f(n) > cg(n) for some positive constant c. Jorge Urrutia [U96] raised the following problem. Is it true that any family ofn pairwise disjoint compact convex sets in the plane has at least f~(n) separable members? In the following special case, the answer is easily seen to be in the affirmative. Let P be a subset of the plane, and let Ha and H2 be the two open half-planes bounded by a straight line g. Cutting P along g, we obtain two pieces/'1 = PDH1 and P2 = PDH2. We say that m pairwise disjoint sets in the plane are separable if we can cut the plane into two parts, and successively cut each part into smaller pieces until we obtain m pieces, each containing precisely one of our m sets. (See Figure 1.) For two positive functions defined on the positive integers , we use the notation f(n) = ~(g(n)) to express that Permission to make digital or hard copies of all or part of this work for personal oz classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the lull citation on the first page. To copy otherwise, to republish, to post on serve1~ or to redistribute to lists, requires prior specific permission and/or a fee. family of n pairwise disjoint compact convex sets in the plane, each containing a circle of radius r and contained in another circle of radius R. Then F has at least cn separable members, where c = c(r,R) > O. cut the plane into squares along …