# On convex polygons contained in convex polygons

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.

• This might be more interesting with a particular example of $n$ and $C$ and meaning for "smaller"
– user44143
Apr 13, 2021 at 2:11
• Is it clear that a smallest region $R$ exists? One can construct a sequence of regions $R_1,R_2,\ldots$ such that for each $i$, the $n$-gon $C$ is the smallest $n$-gon containing $R_i$, but the $R_i$'s limit to a region $R$ such that $C$ is not the 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. May 3, 2021 at 14:30
• That is a possibility I didn't think of. Maybe that may happen if area is the criterion; am unable to figure out. May 4, 2021 at 20:31

No, the region $$R$$ does not have to touch sides of $$C$$ along segments; at least if "smallest" understood in terms of perimeter. Say, for a unit square $$C$$, the inscribed round disc is a solution.
Indeed, in this case, the average width of $$R$$ is at least $$1$$. Therefore perimeter of $$R$$ is at least $$\pi$$.