平衡单位向量
Balancing unit vectors
摘要：
定理A.设$x_1,...,x_{2k+1}$是赋范平面中的单位向量。然后存在标志$\epsi_1,...,\epsi_{2k+1}\in\{\pm 1\}$这样$\norm{\sum_{i=1}^{2k+1}\epsi_i x_i}\leq 1$ .我们利用上述定理的证明方法，给出了如下的点设施选址结果，推广了Y. S. Kupitz和H. Martini（1997）.定理B令$p_0,p_1,...,p_n$是赋范平面中的不同点，使得对于任何$1\leq i<j\leq n$，闭角$\angle p_ip_0p_j$包含与某些$\overrightarrow{p_0p_k}, 1\leq k\leq n$相对的射线。则$p_0$是$\{p_0,p_1,...,p_n\}$的Fermat-Toricelli点，即$x=p_0$最小化$\sum_{i=0}^n\norm{x-p_i}$。我们还证明了定理A的以下动态版本。定理C.令$x_1,x_2,...$是赋范平面中的单位向量序列。然后有存在的迹象$\epsi_1,\epsi_2,...\in\{\pm 1\}$这样$\norm{\sum_{i=1}^{2k}\epsi_i x_i}\leq 2$为所有$k\in\N$。最后，我们讨论了与定理C有关的J. Spencer（1977）的两人平衡对策的一个变形。
英文摘要：
Theorem A. Let $x_1,...,x_{2k+1}$ be unit vectors in a normed plane. Then there exist signs $\epsi_1,...,\epsi_{2k+1}\in\{\pm 1\}$ such that $\norm{\sum_{i=1}^{2k+1}\epsi_i x_i}\leq 1$. We use the method of proof of the above theorem to show the following point facility location result, generalizing Proposition 6.4 of Y. S. Kupitz and H. Martini (1997). Theorem B. Let $p_0,p_1,...,p_n$ be distinct points in a normed plane such that for any $1\leq i<j\leq n$ the closed angle $\angle p_ip_0p_j$ contains a ray opposite some $\overrightarrow{p_0p_k}, 1\leq k\leq n$. Then $p_0$ is a Fermat-Toricelli point of $\{p_0,p_1,...,p_n\}$, i.e. $x=p_0$ minimizes $\sum_{i=0}^n\norm{x-p_i}$. We also prove the following dynamic version of Theorem A. Theorem C. Let $x_1,x_2,...$ be a sequence of unit vectors in a normed plane. Then there exist signs $\epsi_1,\epsi_2,...\in\{\pm 1\}$ such that $\norm{\sum_{i=1}^{2k}\epsi_i x_i}\leq 2$ for all $k\in\N$. Finally we discuss a variation of a two-player balancing game of J. Spencer (1977) related to Theorem C.
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