Balancing unit vectors

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dc.creatorSwanepoel, Konrad J.
dc.date2008-03-04
dc.date.accessioned2026-07-07T09:24:34Z
dc.date.available2026-07-07T09:24:34Z
dc.descriptionTheorem 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.
dc.description7 pages
dc.identifierhttps://arxiv.org/abs/0803.0460
dc.identifierhttp://arxiv.org/abs/0803.0460
dc.identifierJournal of Combinatorial Theory, Series A 89 (2000) 105-112
dc.identifierdoi:10.1006/jcta.1999.3011
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/156144
dc.subjectMetric Geometry
dc.subjectCombinatorics
dc.subject52A40
dc.titleBalancing unit vectors
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