Lower and upper bounds for configurations of points on a sphere

2024/12/08 by Amore, Paolo, Sáenz, Ricardo A.
#FOS: Physical sciences #Mathematical Physics (math-ph) #Soft Condensed Matter (cond-mat.soft)

paper · doi:10.48550/arxiv.2412.05800

Abstract

We present a new proof (based on spectral decomposition) of a bound originally proved by Sidelnikov~ for the frame potentials ∑_ij ( \bf P_i ⋅ \bf P_j )^ℓ on a unit--sphere in d dimensions. Sidelnikov's bound is a special case of the lower bound for the weighted sums ∑_ij f_i f_j ( \bf P_i ⋅ \bf P_j )^ℓ, 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 ∑_ijk^N [ ( \bf P_i × \bf P_j) ⋅ \bf P_k ]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.

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