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

Abstract

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 $f:(0,\infty )\to \mathbb{R}$ be an increasing and continuous function. Define an energy function $$E_n^f(G_n)=\sum _{p<q}f(|A_p{A}_q|),$$ where $|A_p{A}_q|$ 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.

18 pages, 11 figures, accepted by Geometriae Dedicata

References in corpus (1)

• Distributing many points on spheres: minimal energy and designs