TITLE: Representation of vectors in $\mathbb{R}^2$ via differences of small vectors.
QUESTION [15 upvotes]: Is the following fact true?  

Let $v_1,\ldots, v_k \in \mathbb{R}^2$, $\|v_i\|\leq 1$, be vectors that add up to zero. Does there exist a permutation $\sigma\in S_k$ and vectors $w_1,\ldots, w_k \in \mathbb{R}^2$, $\|w_i\|\leq 1$, such that $v_{\sigma(i)}=w_i-w_{i-1}$? (here, I assume $w_0=w_k$)

Edit. Related (known) facts:
1. The same fact in $\mathbb{R}^1$ is true. (can be easily proven by choosing $w_0=0$ and $\sigma(i)$ such that $\|w_i\|\leq 1$ for $w_i:=w_{i-1}+v_{\sigma(i)}$)
2. For each $\epsilon>0$ there exists a family of vectors $v_i$, such that $\max_i\|w_i\|>1-\epsilon$. See my comment below for the proof.

REPLY [8 votes]: I have a counterexample in $\mathbb R^2$.
Here's how it goes.
Pick two numbers $n$ and $N$, with $N>>n>>1$.
The collection {$v_i$} consists of:

$N(n-1)$ times the vector $(\frac{n-2}{n},\frac{1}{N(2n-3)})$
$N(n-2)$ times the vector $(-\frac{n-1}{n},\frac{1}{N(2n-3)})$
The vector $(0,-1)$ once.

The smallest ball into which those vectors can be fit back-to-back has diameter $\sqrt{5}-\varepsilon$.