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 Martínez and Ortega-Moreno. Specifically, for any unit vectors $u_1,\dots ,u_n\in {\mathbb{R}}^d$, there exists a unit vector $v\in {\mathbb{R}}^d$ such that $$\sum _{i=1}^n\frac{1}{⟨u_i,v{⟩}^2}\le n^2.$$ As an immediate consequence, we recover the real linear polarization inequality $$\prod _{i=1}^n|⟨u_i,v⟩|\ge n^{-n/2}.$$

6 pages