I want to find vectors $\textbf{v}_1, \dots,\textbf{v}_n$ in $\mathbb{R}^2$ with that $\sum_{i=1}^n\textbf{v}_i=\textbf{0}$ and $\Vert \textbf{v}_i\Vert\leq 1$ for all $i=1,\dots,n$, such that for every permutation $\pi$ on $\{1,\dots,n\}$, there is some $k=1,\dots,n$ with $\Vert \sum_{i=1}^k \textbf{v}_{\pi(i)}\Vert>1$.
I am reading the Miniature 20 in Matoušek's Thirty-three Miniatures. What I am asking is mentioned by the author on page 70 line 6.
I can only make some trivial observations, say for each such example, we can assume there exists some unit vector $\textbf{u}$ amongst $\textbf{v}_1, \dots,\textbf{v}_n$; If $\textbf{v}$ is one of $\textbf{v}_1, \dots,\textbf{v}_n$, then we can assume $-\textbf{v}$ is not one of them. The author also mentions that there is example that "$\Vert \sum_{i=1}^k \textbf{v}_{\pi(i)}\Vert>1$" can be replaced by "$\Vert \sum_{i=1}^k \textbf{v}_{\pi(i)}\Vert\geq\frac{\sqrt{5}}{2}$", hence, I think we can make some example that "some partial sum must reach $(1,\frac{1}{2})$ for each permutation." Appreciate for any help.