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

Abstract

We prove that given any set of $n$ unit vectors $\{v_i{\}}_{i=1}^n\subset {\mathbb{R}}^n,$ the inequality \[ \sup\limits_{\Vert x \Vert_{\mathbb R^n} =1} \vert \langle x, v_1 \rangle \cdots \langle x, v_n\rangle\vert \ge n^{-n/2} \] holds for $n\le 14.$ Moreover, the equality is attained if and only if $\{v_i{\}}_{i=1}^n$ is an orthonormal system.