An elementary solution of the polarization problem

Abstract

We give a self-contained and entirely elementary proof of the strong polarization inequality: 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 \[ \prod_{i=1}^n |\langle u_i,v\rangle|\ge n^{-n/2}. \]