Finding largest rectangles in convex polygons

We consider the following geometric optimization problem: find a maximum-area rectangle and a maximum-perimeter rectangle contained in a given convex polygon with n vertices. We give exact algorithms that solve these problems in time O(n). We also give (1 − ε)-approximation algorithms that take time O(ε−3/2 + ε−1/2 log n) for maximizing the area and O(ε−3 + ε−1 log n) for maximizing the perimeter.