Minimal positive $2$-spanning sets of vectors

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.