On the real linear polarization constant problem

The present paper deals with lower bounds for the norm of products of linear forms. It has been proved by J. Arias-de-Reyna \cite{ARIAS}, that %for ${\mathbb C}^n$, the so-called $n^{\rm th}$ linear polarization constant $c_n({\mathbb C}^n)$ is $n^{n/2}$, for arbitrary $n\in\NN$. The same value for $c_n({\mathbb R}^n)$ is only conjectured. In a recent work A. Pappas and S. R{\'e}v{\'e}sz prove that $c_n({\mathbb R}^n)=n^{n/2}$ for $n \le 5$. Moreover, they show that if the linear forms are given as $f_j(x)=< x,a_j>$, for some unit vectors $a_j$ $(1\leq j\leq n)$, then the product of the $f_j$'s attains at least the value $n^{-n/2}$ at the normalized signed sum of the vectors $\{a_j\}_{j=1}^{n}$ having maximal length. Thus they asked whether this phenomenon remains true for arbitrary $n\in{\mathbb N}$. We show that for vector systems $\{a_j\}_{j=1}^{n}$ close to an orthonormal system, the Pappas-R{\'e}v{\'e}sz estimate does hold true. Furthermore, among these vector systems the only system giving $n^{-n/2}$ as the norm of the product is the orthonormal system. On the other hand, for arbitrary vector systems we answer the question of A. Pappas and S. R{\'e}v{\'e}sz in the negative when $n\in {\mathbb N}$ is large enough. We also discuss various further examples and counterexamples that may be instructive for further research towards the determination of $c_n(\RR^n)$.