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(n3)O(n3). We also give (1−ε)(1−ε)-approximation algorithms that take time O(ε−1/2log⁡n+ε−3/2)O(ε−1/2log⁡n+ε−3/2).