If $n$ vectors $a\_1, a\_2, \cdots , a\_n$ are given in $\mathbb{R}^n$ with all lengths at most $1$, then it is not to hard to see that we can put $+$ and $-$ in place of $\*$ in the expression
$$a\_1 \* a\_2 \* \cdots \* a\_n$$
so that the result will have length at most $\sqrt{n}$.


To see this, it's enough to choose $\lambda\_1, \cdots, \lambda\_n$ independently and uniformly from $\{\pm 1\}$. Then, put $X={|\sum\_{i=1}^{n}a\_i\lambda\_i|}^2$. By a simple calculation, you will find that $E(X)=n$. Hence there exist specific $\lambda\_1, \cdots, \lambda\_n = \pm 1$ with and with $X\leq n$. Taking 
square roots gives the theorem. 


Now I am curious to know, is it possible to generalize this theorem as follows? 


Assume $N>n$ vectors $a\_1, a\_2, \cdots , a\_N$ are given in $\mathbb{R}^n$ with all lengths at most $1$. Is it possible to put $+$ and $-$ in place of $\*$ in the expression
$$a\_1 \* a\_2 \* \cdots \* a\_N ,$$
so that the result will have length at most $\sqrt{n}$.



Balanced vectors
It is a classical result of [Barany and Grinberg](http://www.sciencedirect.com/science/article/pii/0024379581900859) (generalizing an earlier result of Spencer) that there exist $\lambda\_1,\dotsc,\lambda\_N\in\{\pm 1\}$ with
 $$ \|\lambda\_1a\_1+\dotsb+\lambda\_Na\_N\| \le 2n. $$
The paper of Barany-Grinberg was published in 1981, but they indicated that the problem was posed in 1963 by Dvoretzky. Interestingly, Barany and Grinberg, along with Gergely Ambrus, have just published [another paper](https://arxiv.org/abs/1502.04027) on this subject.


Notice that for $n=2$, it is easy to get the sharp bound
 $$ \|\lambda\_1a\_1+\dotsb+\lambda\_Na\_N\| \le \sqrt 2; $$
it follows by using induction and observing that among any three vectors in $\mathbb R^2$ of length at most $1$, there are two vectors such that either their sum, or their difference has length at most $1$, and that for any two vectors of length at most $1$, either their sum, or their difference has length at most $\sqrt 2$.


