A lower bound for the logarithmic energy on $\mathbb{S}^2$ and for the Green energy on $\mathbb{S}^n$

TL;DR

This paper provides an alternative proof for the best known lower bound of logarithmic energy on the sphere and extends the method to establish new bounds for Green energy on higher-dimensional spheres.

Contribution

It introduces a novel proof technique for energy bounds and generalizes it from two-dimensional to n-dimensional spheres, advancing theoretical understanding.

Findings

Sharpest known lower bound for logarithmic energy on -sphere

New lower bounds for Green energy on -spheres

Generalization of proof method to higher dimensions

Abstract

We show an alternative proof of the sharpest known lower bound for the logarithmic energy on the unit sphere $\mathbb{S}^2$. We then generalize this proof to get new lower bounds for the Green energy on the unit $n$-sphere $\mathbb{S}^n$.