An elementary solution of the polarization problem

Abstract

By distilling and combining earlier arguments, we give a self-contained and entirely elementary proof of the strong and real linear polarization inequalities established by Mart\'inez and Ortega-Moreno. Specifically, for any unit vectors $u_1,\ldots,u_n\in\mathbb{R}^d$, there exists a unit vector $v\in\mathbb{R}^d$ such that $$\sum_{i=1}^n \frac{1}{\langle u_i,v\rangle^2}\le n^2. $$ As an immediate consequence, we recover the real linear polarization inequality $$\prod_{i=1}^n |\langle u_i,v\rangle|\ge n^{-n/2}. $$