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arXiv · 2609.31124

An elementary solution of the polarization problem

Gergely Ambrus
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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 $u_1,\ldots,u_n\in\mathbb{R}^d$, 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 $$\prod_{i=1}^n |\langle u_i,v\rangle|\ge n^{-n/2}. $$

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Gergely Ambrus. 2026-10-03. An elementary solution of the polarization problem. https://arxiv.org/abs/2609.31124

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