The authors consider upper bounds on the number of intersections between two rectangular paths. Let these two paths be denoted as P and Q, and denote the number of Manhattan subpaths in P and Q by mod P mod and mod Q mod respectively. K. Kant (1985) gave an upper bound of 10 mod P mod mod Q mod /9+4( mod P mod + mod Q mod )/9. The authors have sharpened these upper bounds, using methods to break the rectangular paths into subpaths, to be mod P mod mod Q mod +( mod P mod /2)+( mod Q mod /3), where they assume without loss of generality that mod P mod>

We give a finiteness criteria for the potato-peeling problem that asks for the largest convex Polygon ('Potato') contained inside a given simple polygon, answering a question of J. Goodman. This leads to a polynomial-time, solution of O(n/sup 9/log n). The techniques used turn out to be useful for other cases of what we call the polygon inclusion and enclosure problem. For instance, the largest perimeter potato can be found in O(n/sup 6/) time and finding the smallest k-gon enclosing a given polygon can be done in O(n/sup 3/log k) steps.