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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 u1,…,un∈ℝd, there exists a unit vector v∈ℝd such that ∑i=1n 1/(⟨ ui,v⟩2)≤ n2. As an immediate consequence, we recover the real linear polarization inequality ∏i=1n |⟨ ui,v⟩|≥ n−n/2.
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BibTeX
@misc{ambrus2026elementary,
title = {{An elementary solution of the polarization problem}},
author = {Gergely Ambrus},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.31124},
eprint = {2609.31124},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31124},
}
RIS
TY - UNPB
TI - An elementary solution of the polarization problem
AU - Gergely Ambrus
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2609.31124
UR - https://arxiv.org/abs/2609.31124
AB - 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,…,u_n∈ℝ^d, there exists a unit vector v∈ℝ^d such that ∑_i=1^n 1/(⟨ u_i,v⟩^2)≤ n^2. As an immediate consequence, we recover the real linear polarization inequality ∏_i=1^n |⟨ u_i,v⟩|≥ n^−n/2.
ER -