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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Minimal positive $ 2$-spanning sets of vectors

Author: Daniel A. Marcus
Journal: Proc. Amer. Math. Soc. 82 (1981), 165-172
MSC: Primary 15A03; Secondary 52A25
MathSciNet review: 609644
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Abstract: Let $ ({\upsilon _1}, \ldots ,{\upsilon _n})$ be a sequence in an $ m$-dimensional vector space $ V$ over an ordered field such that, for each $ i$, $ \left\{ {{\upsilon _j}:j \ne i} \right\}$ positively spans $ V$. It is shown that if $ ({\upsilon _1}, \ldots ,{\upsilon _n})$ is minimal with this property, then

$\displaystyle n \leqslant \left\{ {_{m(m + 1)/2 + 5}^{4m}} \right.\quad _{{\text{if}}\;m \geqslant 5}^{{\text{if}}\;m \leqslant 5}$

and all cases are determined in which $ n = 4m$, $ m \leqslant 4$. An application to convex polytopes is given.

References [Enhancements On Off] (What's this?)

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  • [2] C. Davis, Theory of positive linear dependence, Amer. J. Math. 76 (1954), 733-746. MR 0064011 (16:211e)
  • [3] B. Grünbaum, Convex polytopes, Interscience, New York, 1967.
  • [4] D. Marcus, Gale diagrams of convex polytopes and positive spanning sets of vectors (to appear). MR 754428 (85j:52016)
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Article copyright: © Copyright 1981 American Mathematical Society

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