A finiteness criterion for the potato-peeling problem is given 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**9 log n). The techniques used turn out to be useful for other cases of what are called the polygon inclusion and enclosure problems. For instance, the largest perimeter potato can be found in O(n**6 ) time, and finding the smallest k-gon enclosing a given polygon can be done in O(n**3 log k) steps.