I have a counterexample in $\mathbb R^2$. Here's how it goes.<br> Pick two numbers $n$ and $N$, with $N>>n>>1$.<br> 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 $\sqrt{5}-\varepsilon$.