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 $m$.)
