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Published online by Cambridge University Press: 05 December 2014
Given a finite-dimensional Banach space 
$X$ and an Auerbach basis 
$\{(x_{k},x_{k}^{\ast }):1\leqslant k\leqslant n\}$ of 
$X$, it is proved that there exist 
$n+1$ linear combinations 
$z_{1},\ldots ,z_{n+1}$ of 
$x_{1},\ldots ,x_{n}$ with coordinates 
$0,\pm 1$, such that 
$\Vert z_{k}\Vert =1$, for 
$k=1$, 
$2,\ldots ,n+1$ and 
$\Vert z_{k}-z_{l}\Vert >1$, for 
$1\leqslant k<l\leqslant n+1$.
