# Optimal measures for p-frame energies on spheres

## TL;DR

This paper investigates optimal arrangements of points on spheres to minimize p-frame energies, using linear programming and symmetry, revealing new optimal configurations and bounds, especially for non-even p values.

## Contribution

It introduces a comprehensive method to identify optimal measures for p-frame energies on spheres, including new results for tight designs and the 600-cell, and extends to broader potential functions.

## Key findings

- Optimality of tight designs and the 600-cell for certain p ranges.
- Application of linear programming to describe minimizing measures.
- Improved lower bounds for weighted designs in complex projective spaces.

## Abstract

We provide new answers about the placement of mass on spheres so as to minimize energies of pairwise interactions. We find optimal measures for the $p$-frame energies, i.e. energies with the kernel given by the absolute value of the inner product raised to a positive power $p$. Application of linear programming methods in the setting of projective spaces allows for describing the minimizing measures in full in several cases: we show optimality of tight designs and of the $600$-cell for several ranges of $p$ in different dimensions. Our methods apply to a much broader class of potential functions, those which are absolutely monotonic up to a particular order as functions of the cosine of the geodesic distance. In addition, a preliminary numerical study is presented which suggests optimality of several other highly symmetric configurations and weighted designs in low dimensions. In one case we improve the best known lower bounds on a minimal sized weighted design in $\mathbb{CP}^4$. All these results point to the discreteness of minimizing measures for the $p$-frame energy with $p$ not an even integer.