Minimal positive $2$-spanning sets of vectors
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- by Daniel A. Marcus PDF
- Proc. Amer. Math. Soc. 82 (1981), 165-172 Request permission
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 \[ n \leqslant \left \{ {_{m(m + 1)/2 + 5}^{4m}} \right .\quad _{{\text {if}}\;m \geqslant 5}^{{\text {if}}\;m \leqslant 5}\] and all cases are determined in which $n = 4m$, $m \leqslant 4$. An application to convex polytopes is given.References
- Michel Dalmazzo, Nombre d’arcs dans les graphes $k$-arc-fortement connexes minimaux, C. R. Acad. Sci. Paris Sér. A-B 285 (1977), no. 5, A341–A344 (French, with English summary). MR 463004
- Chandler Davis, Theory of positive linear dependence, Amer. J. Math. 76 (1954), 733–746. MR 64011, DOI 10.2307/2372648 B. Grünbaum, Convex polytopes, Interscience, New York, 1967.
- Daniel A. Marcus, Gale diagrams of convex polytopes and positive spanning sets of vectors, Discrete Appl. Math. 9 (1984), no. 1, 47–67. MR 754428, DOI 10.1016/0166-218X(84)90090-8 —, Circulation polytopes associated with directed graphs (to appear). P. McMullen, Representations and diagrams (to appear).
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 82 (1981), 165-172
- MSC: Primary 15A03; Secondary 52A25
- DOI: https://doi.org/10.1090/S0002-9939-1981-0609644-3
- MathSciNet review: 609644