发表机构
Universidad Torcuato Di Tella; CONICET（托尔夸托·迪泰拉大学; 阿根廷国家科学研究委员会）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究确定了实极化问题中Bang向量满足极化不等式的精确维数范围，证明n≤14时规则普遍有效，n≥15时存在反例，为该领域提供了关键的维数边界结论。
AI 中文摘要
我们确定了归一化最长符号和（称为Bang向量）始终满足实极化不等式的精确维数范围：当且仅当n≤14时，该规则普遍有效。对于每个n≥15，我们构造了n个单位向量，其全正和在整体符号下是唯一的最长符号和，而其归一化方向的极化乘积小于n^{-n/2}。反例由一个显式族给出，对于相同构型，我们展示了非最大符号和，其归一化方向确实满足极化界。
英文摘要
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 \(n\leq 14\). For every \(n\geq 15\) 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.
Comments11 pages
