Balancing unit vectors

Abstract

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 $1\le i<j\le n$ the closed angle $\angle p_i{p}_0{p}_j$ contains a ray opposite some $\overrightarrow{p_0{p}_k},1\le k\le n$. Then $p_0$ is a Fermat-Toricelli point of $\{p_0,p_1,...,p_n\}$, 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 $k\in \mathbb{N}$. Finally we discuss a variation of a two-player balancing game of J. Spencer (1977) related to Theorem C.

7 pages

Cited by in corpus (1)

• The Fermat-Torricelli problem in normed planes and spaces