This question just popped into my head while doing some "for fun" math.
More precisely:
Let $m,n\in\Bbb{Z};m,n>2$. Let $P$ be a regular $n$-gon (let's say $P$ is the convex hull of the $n$ $n$th roots of unity in $\Bbb{C}$). Let $\mathcal{Q}$ be the set of all regular $m$-gons $Q$ in $\Bbb{C}$ such that $Q\subseteq P$. Is there a known method for determining $\sup_{Q\in\mathcal{Q}}|Q|$ in terms of $m$ and $n$? ($|Q|$=the area of $Q$.)
Further questions: What is $\inf_{m,n>2}\frac{\sup_{Q\in\mathcal{Q}}|Q|}{|P|}$? Is a minimum actually achieved for some pair $(m,n)$? I feel like the answer to the latter question is $(3,4)$, i.e. triangle-in-square $($which implies an answer of $2\sqrt{3}-3\approx0.464$ to the former question$)$.
Apologies if this has been asked before! This is my first post here.