# On the linear polarization constants of finite dimensional spaces

## TL;DR

This paper investigates the asymptotic behavior of linear polarization constants in finite-dimensional Banach spaces, providing precise growth rates for spaces ill_p^d across different p-values.

## Contribution

It determines the exact asymptotic behavior of linear polarization constants for ill_p^d spaces, clarifying their growth rates for all p between 1 and infinity.

## Key findings

- Constants grow as  d^{1/p} for 1  p  2
- Constants grow as   d^{1/2} for 2  p <   infinity
- For p= infinity, asymptotic behavior is established up to a logarithmic factor

## Abstract

We study the linear polarization constants of finite dimensional Banach spaces. We obtain the correct asymptotic behaviour of these constants for the spaces $\ell_p^d$: they behave as $\sqrt[p]{d}$ if $1\le p\le 2$ and as $\sqrt{d}$ if $2\le p<\infty$. For $p=\infty$ we get the asymptotic behavior up to a logarithmic factor