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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.

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  • $\begingroup$ This might be more interesting with a particular example of $n$ and $C$ and meaning for "smaller" $\endgroup$
    – user44143
    Apr 13, 2021 at 2:11
  • $\begingroup$ 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. $\endgroup$
    – Nathan Reading
    May 3, 2021 at 14:30
  • $\begingroup$ That is a possibility I didn't think of. Maybe that may happen if area is the criterion; am unable to figure out. $\endgroup$
    – Nandakumar R
    May 4, 2021 at 20:31

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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 $\pi$.

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  • $\begingroup$ Thanks. I understand this implies "the smallest perimeter convex region R for which the unit square is the least perimeter 4-gon containing it is the unit disc". Hope you could explain the observation that the average width of R has to be at least 1. $\endgroup$
    – Nandakumar R
    May 4, 2021 at 20:28
  • $\begingroup$ @NandakumarR Consider all circumscribed rectangles. Its perimeter is 4 or more. So average widths in two orthogonal directions have to be at least 1. $\endgroup$
    – Anton Petrunin
    May 4, 2021 at 20:32

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