On the $n-$th linear polarization constant of ${\mathbb{R}}^n$

We prove that given any set of $n$ unit vectors $\{v_i{\}}_{i=1}^n\subset {\mathbb{R}}^n,$ the inequality $$\underset{\mathrm{\parallel }x{\mathrm{\parallel }}_{{\mathbb{R}}^n}=1}{\sup }\mathrm{\mid }⟨x,v_1⟩\cdots ⟨x,v_n⟩\mathrm{\mid }\ge n^{-n\mathrm{/}2}$$ holds for $n\le 14.$ Moreover, the equality is attained if and only if $\{v_i{\}}_{i=1}^n$ is an orthonormal system.