Balanced vectors

It is a classical result of Barany and Grinberg (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 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 $\sqrt 2$.