  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 $O(\varepsilon^{-3/2}+ \varepsilon^{-1/2} \log n)$.
