A discrete isoperimetric problem (Q674793)

Let \(\pi (n)\) be the maximum perimeter of a convex \(n\)-gon of diameter \(1\). \textit{D. G. Larman} and \textit{N. K. Tamvakis} [Ann. Discrete Math. 20, 209-214 (1984; Zbl 0565.52002)] have proved that \(\pi (n)=2n\sin (\pi /2n)\) whenever \(n\) is not a power of \(2\). The author shows that \(\pi (n)\leq 2n\sin (\pi /2n)\) for arbitrary \(n\geq 3\) with equality only if \(n\) is not a power of \(2\). For the latter case, he gives an explicit construction of all extremal \(n\)-gons based on a one-to-one correspondence with the solutions of a certain diophantine problem. Each of these extremal \(n\)-gons has equal sides and is inscribed in a Reuleaux \(l\)-gon for an odd integer \(l\), \(1<l\leq n\). (As the author remarks, this property was proved already by Reinhardt in 1922.) For \(n\) a power of 2, the problem is solved completely only for \(n=4\).