AI 中文总结
该文综合前人思想，给出强极化与线性极化问题的自包含初等证明。
AI 中文摘要
通过提炼并综合以往论证中的思想，我们提供了强极化问题和线性极化问题的一个自包含且完全初等的证明。
英文摘要
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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