Lower and upper bounds for configurations of points on a sphere

Abstract

We present a new proof (based on spectral decomposition) of a bound originally proved by Sidelnikov~\, for the frame potentials ${\sum }_{ij}{\left({\mathbf{P}}_i\cdot {\mathbf{P}}_j\right)}^{\ell }$ on a unit--sphere in $d$ dimensions. Sidelnikov's bound is a special case of the lower bound for the weighted sums ${\sum }_{ij}{f}_i{f}_j{\left({\mathbf{P}}_i\cdot {\mathbf{P}}_j\right)}^{\ell }$, where $f_i>0$ are scalar quantities associated to each point on the sphere, which we also prove using spectral decomposition. Moreover, in three dimensions, again using spectral decomposition, we find a sharp upper bound for ${\sum }_{ijk}^N{\left[\left({\mathbf{P}}_i\times {\mathbf{P}}_j\right)\cdot {\mathbf{P}}_k\right]}^2$. We explore two applications of these bounds: first, we examine configurations of points corresponding to the local minima of the Thomson problem for $N=972$; second, we analyze various distributions of points within a three-dimensional volume, where a suitable weighted sum is defined to satisfy a specific bound.

22 pages, 6 figures