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.

notthe smallest $n$-gon containing $R$. (Presumably, $R$ would be an $n$-gon with one vertex on each edge of $C$.) I could imagine that such a sequence approaches the infimum of "size" (in whatever sense we want to say "smallest") and that no smallest $R$ exists. $\endgroup$areais the criterion; am unable to figure out. $\endgroup$