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 $O(n^3)$. We also give $(1-\varepsilon)$-approximation algorithms that take time $O(\varepsilon^{-3/2}+ \varepsilon^{-1/2} \log n)$. 
Sergio Cabello, Otfried Cheong, Christian Knauer, Lena Schlipf
cs.CG