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

TL;DR

This paper establishes a lower bound for the maximum product of inner products between a unit vector and a set of vectors in 0^n for dimensions up to 14, with equality characterizing orthonormal systems.

Contribution

It proves a specific inequality for the n-th linear polarization constant in 0^n for n 0 14, identifying conditions for equality.

Findings

Lower bound of n^{-n/2} for the product of inner products

Equality holds if and only if vectors form an orthonormal system

Valid for dimensions up to 14

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.