The O’Hara conjecture on maximum energies of equilateral convex polygons

Let $\Gamma_n$ denote the set of convex $n$-gons with each edge of length $1$, and let $E_n^\alpha$ be the energy associated with the increasing continuous function $f_\alpha:(0,\infty)\to\mathbb R$. For even $n>0$, let $\Gamma_n^-$ denote the double straight arc, defined by a degenerate polygon whose one diagonal has length $n/2$. O’Hara conjecture. There is a constant $\alpha_n^*>0$ such that, for $\alpha>\alpha_n^*$, $E_n^\alpha(\Gamma_n)$ reaches its maximum if and only if $\Gamma_n$ is the double straight arc $\Gamma_n^-$. The conjecture concerns the shape of equilateral convex polygons maximizing the energy; it was verified by Abrams, Cantarella, Fu, Ghomi, and Howard.