Discrete isoperimetric inequality involving the diameter of an n-gon

Question

I am interested in discrete isoperimetric-type inequalities that allow one to bound the perimeter of an $n$-gon from above (as opposed to below, as in the classical case when one bounds the perimeter by the square root of its area times a constant dependent on $n$).

If $P$ is a $n$-gon and $\operatorname{Diam}(P)$ is the diameter of $P$ (i.e., the largest distance attained by any two points of $P$), then I conjecture that the inequality

$$\frac{\operatorname{Perimeter}(P)}{\operatorname{Diam}(P)} \leq n\sin\frac{\pi}{n}\csc\frac{(n - \delta_n)\pi}{2n}$$

must hold true, with equality holding if and only if $P$ is a regular $n$-gon. Here $\delta_n = 1$ if $n$ is odd and $\delta_n = 0$ if $n$ is even. Regardless of whether this is true, this ratio must have been well-studied. Can you please share a reference?

It could also be that, instead of a diameter, it is more natural to bound the perimeter from above in terms of some other quantity associated with $P$. If so, what quantity would be more natural to consider, and where can I find the proof of the associated discrete isoperimetric inequality?

P.S. I would also appreciate a reference where I can find the proof of the inequality $4n\tan\frac{\pi}{n}\leq \frac{\operatorname{Perimeter}^2(P)}{\operatorname{Area}(P)}$, which allows one to bound the perimeter of $P$ from below.

Answer 1

According to a survey by Tóth, an isoperimetric inequality for convex $n$-gons was given by Reinhardt in 1922, who showed that $$ \frac{\text{perimeter}(P)}{\text{diameter}(P)} \leq 2n \sin\left(\frac{\pi}{2n}\right), $$ with equality attained if and only if $n$ is not a power of $2$. The polygons which maximize the perimeter for a given diameter are known as Reinhardt polygons. For $n = 2^k$, sharp bounds are only known for $k \leq 3$ as of 2022.

See Tóth, Gábor Fejes. “Finite variations on the isoperimetric problem.” (2022).