# Frame Potentials and Orthogonal Vectors

## TL;DR

The paper extends a recent result by Glazyrin, identifying configurations of vectors that minimize the p-frame potential for certain parameters, specifically involving orthonormal bases combined with additional vectors.

## Contribution

It generalizes the minimization of the p-frame potential to include orthonormal bases combined with extra vectors for specific p ranges.

## Key findings

- Orthonormal basis plus additional vectors minimize the p-frame potential within the specified range.
- The result extends previous work by Glazyrin to a broader class of vector configurations.
- Provides explicit conditions for minimal configurations in high-dimensional spheres.

## Abstract

An extension is given of a recent result of Glazyrin, showing that an orthonormal basis $\{e_{i}\}_{i=1}^{d}$ joined with the vectors $\{e_{j}\}_{j=1}^{m}$, where $1\leq m < d$ minimizes the $p$-frame potential for $p\in[1,2\log{\frac{2m+1}{2m}}/\log{\frac{m+1}{m}}]$ over all collections of $N=d+m$ vectors $\{x_1,\dots,x_N \}$ in $\mathbb{S}^{d-1}$.