On the linear polarization constants of finite dimensional spaces

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