Largest inscribed rectangles in convex polygons

We consider approximation algorithms for the problem of computing an inscribed rectangle having largest area in a convex polygon on n vertices. If the order of the vertices of the polygon is given, we present a randomized algorithm that computes an inscribed rectangle with area at least (1− ) times the optimum with probability t in time O( 1 log n) for any constant t < 1. We further give a deterministic approximation algorithm that computes an inscribed rectangle of area at least (1− ) times the optimum in running time O( 1 2 log n) and show how this running time can be slightly improved.