I have proved a simple Lemma that I need for a larger result, and I was wondering whether it is actually another more famous result in disguise.


The lemma says that for any set of vectors in $\mathbb{R}^n$, it is possible to choose a set of coefficients in ${-1,+1}$ such that the weighted sum of the vectors is close to the origin. The interesting thing (to me, non-mathematician) is that how close depends on the dimensionality of the space and not on the number of vectors.


**Lemma** Consider a set of $m\geq1$ vectors $\{v\_{i}\}\_{i=1}^{m}$ in $\mathbb{R}^{n}$, $n\geq1$, such that $||v\_{i}||\_{\infty}\leq c$. Then there exists a vector of coefficients $\{x\_{i}\}\_{i=1}^{m}$ , $x\_{i}\in\{-1,+1\}$ , such that $\left\Vert \sum\_{i=1}^{m}x\_{i}v\_{i}\right\Vert \leq nc$.


(Note that the bound depends on the dimension $n$, but not the number of vectors $m$.)



More general form of inequality?
I don't have access right now to search tools, but the result is is due to Steinitz and might be described in 


Kadets, Mikhail I.; Kadets, Vladimir M. (1997). "Chapter 2.1 Steinitz's theorem on the sum range of a series, Chapter 7 The Steinitz theorem and B-convexity". Series in Banach spaces: Conditional and unconditional convergence. Operator Theory: Advances and Applications. 94.


Anyway, the right book by V. Kadets contains the result and best known estimates for the constant in terms of the dimension.


EDIT: I asked Gideon Schechtman for a good reference, and he provided


[this](http://scholar.google.co.il/scholar_url?hl=iw&q=http://citeseerx.ist.psu.edu/viewdoc/download%3Fdoi%3D10.1.1.145.1357%26rep%3Drep1%26type%3Dpdf&sa=X&scisig=AAGBfm1YEK9BPZT-XOGtAeF5YY8WmbszDA&oi=scholarr&ei=YfeBT5-GMcSm4gTU9pjvBw&ved=0CBoQgAMoADAA)


