# Isoperimetric inequality inside a regular polygon

Let $\mathcal{P}_n$ be a fixed $n$-sided regular polygon with area $A:=\vert \mathcal{P}_n\vert>0$. For any $c\in (0,A)$, I would like to find the shape of the domain $D\subset \mathcal{P}_n$ such that $\vert D \vert=c$ and the perimeter $\vert\partial D \vert$ is minimal. Do you have an idea how to tackle the problem or is there any related work you know of?

Of course, for small $c$, we can choose $D$ to be a disc because it is a global solution to the isoperimetric problem. But what happens if $c$ is too large such that the corresponding disc can not stay within the polygon?

Best wishes

• locally the boundary must be circular and then touch the sides tangentially. – user35593 Aug 11 '18 at 13:04
• Interesting! That is what I had expected. Do you know any reference or a proof which makes that idea precise? – Oliver Watt Aug 11 '18 at 13:30
• Is it your homework? – Anton Petrunin Aug 11 '18 at 22:38
• No, it is not my homework. – Oliver Watt Aug 12 '18 at 5:44