Distributions of points on non-extensible closed curves in ${\mathbb{R}}^3$ realizing maximum energies

Let $G_n$ be a non-extensible, flexible closed curve of length $n$ in the 3-space ${\mathbb{R}}^3$ with $n$ particles $A_1$,…,$A_n$ evenly fixed (according to the arc length of $G_n$) on the curve. Let $f:(0,\mathrm{\infty })\to \mathbb{R}$ be an increasing and continuous function. Define an energy function $$E_n^f(G_n)=\sum _{p<q}f(\mathrm{\mid }{A}_p{A}_q\mathrm{\mid }),$$ where $\mathrm{\mid }{A}_p{A}_q\mathrm{\mid }$ is the distance between $A_p$ and $A_q$ in ${\mathbb{R}}^3$. We address a natural and interesting problem: {\it What is the shape of $G_n$ when $E_n^f(G_n)$ reaches the maximum? } In many natural cases, one such case being $f(t)=t^{\alpha }$ with $0<\alpha \le 2$, the maximizers are regular $n$-gons and in all cases the maximizers are (possibly degenerate) convex $n$-gons with each edge of length 1.