单位向量的大幅符号和：首个线性相关情形
Large signed sums of unit vectors: the first linearly dependent case
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中文总结 AI 辅助
本文解决了单位向量大幅符号和的最小最大值问题，证明在$d+1$个$d$维单位向量的情况下该值为$\sqrt{d+2}$，并刻画了所有等号情形。
中文摘要 AI 辅助
我们研究了一个关于单位向量大幅符号和的问题，该问题源于Brugger、Fiedler、González Merino和Kirschbaum的工作，后来由Ambrus和Nietert以当前形式提出。给定$\mathbb R^d$中的$d+1$个单位向量$u_1,\ldots,u_{d+1}$，其中$d\ge2$，问题要求确定\\[ \max_{\varepsilon_i=\pm1} \left\\|\sum_{i=1}^{d+1}\varepsilon_i u_i\right\\| \\]的最小可能值。我们证明该值为$\sqrt{d+2}$。我们还确定了所有等号情形：在独立的符号变化和正交变换下，它们由正偶数维中心正则单纯形的顶点及其正交补空间的一组标准正交基组成。
英文摘要
We address a problem on large signed sums of unit vectors that arose in work of Brugger, Fiedler, González Merino and Kirschbaum and was later formulated in its present form by Ambrus and Nietert. Given $d+1$ unit vectors $u_1,\ldots,u_{d+1}$ in $\mathbb R^d$, with $d\ge2$, the problem asks for the smallest possible value of \[ \max_{\varepsilon_i=\pm1} \left\|\sum_{i=1}^{d+1}\varepsilon_i u_i\right\|. \] We prove that this value is $\sqrt{d+2}$. We also determine all equality cases: up to independent sign changes and orthogonal transformations, they consist of the vertices of a centered regular simplex of positive even dimension together with an orthonormal basis of its orthogonal complement.
