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 $2$. Equality is obtained for so called 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).

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).