Balancing unit vectors

Theorem A. Let $x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_{2k+1}$ be unit vectors in a normed plane. Then there exist signs ${\text{epsi}}_1,\mathrm{.}\mathrm{.}\mathrm{.},{\text{epsi}}_{2k+1}\in \{\pm 1\}$ such that $\text{norm}{\sum }_{i=1}^{2k+1}{\text{epsi}}_i{x}_i\le 1$. 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 $p_0,p_1,\mathrm{.}\mathrm{.}\mathrm{.},p_n$ be distinct points in a normed plane such that for any $1\le i<j\le n$ the closed angle $\mathrm{\angle }{p}_i{p}_0{p}_j$ contains a ray opposite some $\overrightarrow{p_0{p}_k},1\le k\le n$. Then $p_0$ is a Fermat-Toricelli point of $\{p_0,p_1,\mathrm{.}\mathrm{.}\mathrm{.},p_n\}$, i.e. $x=p_0$ minimizes ${\sum }_{i=0}^n\text{norm}x-p_i$. We also prove the following dynamic version of Theorem A. Theorem C. Let $x_1,x_2,\mathrm{.}\mathrm{.}\mathrm{.}$ be a sequence of unit vectors in a normed plane. Then there exist signs ${\text{epsi}}_1,{\text{epsi}}_2,\mathrm{.}\mathrm{.}\mathrm{.}\in \{\pm 1\}$ such that $\text{norm}{\sum }_{i=1}^{2k}{\text{epsi}}_i{x}_i\le 2$ for all $k\in \mathbb{N}$. Finally we discuss a variation of a two-player balancing game of J. Spencer (1977) related to Theorem C.