# Polarization, sign sequences and isotropic vector systems

## TL;DR

This paper investigates the polarization constants of the unit sphere across different p-values, identifying extremizers and optimal configurations, with specific results for p=1, p=2, and the circle case.

## Contribution

It determines the order of magnitude of the polarization constants for all dimensions and p-values, and characterizes extremizers for p=2 and p=1, including the case of the circle.

## Key findings

- Extremizers for p=2 are isotropic vector sets.
- Polarization problem for p=1 reduces to maximizing signed vector sums.
- Equally spaced configurations are optimal on the circle.

## Abstract

We determine the order of magnitude of the $n$th $\ell_p$-polarization constant of the unit sphere $S^{d-1}$ for every $n,d \geq 1$ and $p>0$. For $p=2$, we prove that extremizers are isotropic vector sets, whereas for $p=1$, we show that the polarization problem is equivalent to that of maximizing the norm of signed vector sums. Finally, for $d=2$, we discuss the optimality of equally spaced configurations on the unit circle.