Sums of Squared Distances between Points on a Unit $n$-Sphere

TL;DR

This paper generalizes known facts about sums of squared distances on regular polygons to points on a unit n-sphere, providing new theorems that relate distances, centroids, and symmetries, with applications to potential energy calculations.

Contribution

The paper introduces two theorems that extend classical distance sum results from polygons to points on a unit n-sphere, incorporating centroid and symmetry considerations.

Findings

Sum of squared distances depends on centroid distance from sphere center.

Sum of squared distinct distances is determined by the number of distinct distances and symmetry.

New method to compute potential energy of finite normalized frames.

Abstract

In this paper, we prove two theorems concerning the sums of squared distances between points on a unit $n$-sphere that generalize two facts previously known about the case where the points are the vertices of a regular polygon. The first theorem is that, given a multiset of $V$ points on a unit $n$-sphere, the sum of the squared distances between these points is $V^2 ( 1 - d^2 )$ where $d$ is the distance between the centroid of the points and the center of the unit $n$-sphere (for any $n \geq 2$). The second is that, given a finite set of points on the unit $n$-sphere centered at the origin such that the point set is symmetric about the origin and the symmetry group of the point set acts transitively on the set, the sum of the squared distinct distances between these points is $2k + 2$ where $k$ is the number of distinct distances between the points (for any $n \geq 2$). Using the first theorem, we find a new way to calculate the potential energy function of a finite normalized frame.