PreprintarXiv licencearXiv
Bang vectors in the real polarization problem: The exact dimensional range
Abstract
We determine the exact dimensional range in which a normalized longest signed sum, which we call a Bang vector, always satisfies the real polarization inequality: this prescription is universally valid if and only if n≤ 14. For every n≥ 15 we construct n unit vectors whose all-positive sum is, up to global sign, the unique longest signed sum, while its normalized direction has polarization product smaller than n−n/2. The counterexamples are given by a single explicit family, and for the same configurations we exhibit nonmaximal signed sums whose normalized directions do satisfy the polarization bound.
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BibTeX
@misc{pinasco2026bang,
title = {{Bang vectors in the real polarization problem: The exact dimensional range}},
author = {Damián Pinasco},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.31101},
eprint = {2608.31101},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.31101},
}
RIS
TY - UNPB
TI - Bang vectors in the real polarization problem: The exact dimensional range
AU - Damián Pinasco
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.31101
UR - https://arxiv.org/abs/2608.31101
AB - We determine the exact dimensional range in which a normalized longest signed sum, which we call a Bang vector, always satisfies the real polarization inequality: this prescription is universally valid if and only if n≤ 14. For every n≥ 15 we construct n unit vectors whose all-positive sum is, up to global sign, the unique longest signed sum, while its normalized direction has polarization product smaller than n^−n/2. The counterexamples are given by a single explicit family, and for the same configurations we exhibit nonmaximal signed sums whose normalized directions do satisfy the polarization bound.
ER -