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

Abstract

Let $G_n$ be a non-extensible, flexible closed curve of length $n$ in the 3-space $\R^3$ with $n$ particles $A_1$,...,$A_n$ evenly fixed (according to the arc length of $G_n$) on the curve. Let $f:(0, \infty)\to \R$ be an increasing and continuous function. Define an energy function $$E^f_n(G_n)= \sum_{p< q} f(|A_pA_q|),$$ where $|A_pA_q|$ is the distance between $A_p$ and $A_q$ in $\R^3$. We address a natural and interesting problem: {\it What is the shape of $G_n$ when $E^f_n(G_n)$ 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.