In what follows '$n$-gon' stands for '$n$-vertex polygonal region'.
Question: Given a convex $n$-gon $C$, find the smallest convex region $R$ such that $C$ is the smallest $n$-gon that contains it.
Remarks: the 2 'smallest' can independently mean either of 'least area' or 'least perimeter' thus we have 2 questions - indeed, since the two 'smallest's in the question are independent, there are 2 further 'mixed' questions which seem less intuitive. It seems that for both (or may be all of these) questions, for any $C$, $R$ has to touch every side of $n$ along a segment - and thus $R$ should have $2n$ vertices. Another question which one can ask is if for any $C$, all or some of the 4 questions have the same $R$ as answer.
Note: If 'smallest' is given other meanings, say 'smallest diameter', there are even more questions in there.