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.