Extremal convex polygons inscribed in a given convex polygon

A convex polygon $Q$ is inscribed in a convex polygon $P$ if every side of $P$ contains at least one vertex of $Q$. We present algorithms for finding a minimum area and a minimum perimeter convex polygon inscribed in any given convex $n$-gon in $O(n)$ and $O(n^3)$ time, respectively. We also investigate other variants of this problem.