Are there five complex numbers satisfying the following equalities? Can anyone help on the following question? 
Are there five complex numbers $z_{1}$, $z_{2}$ , $z_{3}$ , $z_{4}$
and $z_{5}$ with $\left|z_{1}\right|+\left|z_{2}\right|+\left|z_{3}\right|+\left|z_{4}\right|+\left|z_{5}\right|=1$
such that the smallest among $\left|z_{1}\right|+\left|z_{2}\right|-\left|z_{1}+z_{2}\right|$,
$\left|z_{1}\right|+\left|z_{3}\right|-\left|z_{1}+z_{3}\right|$,
$\left|z_{1}\right|+\left|z_{4}\right|-\left|z_{1}+z_{4}\right|$,
$\left|z_{1}\right|+\left|z_{5}\right|-\left|z_{1}+z_{5}\right|$,
$\left|z_{2}\right|+\left|z_{3}\right|-\left|z_{2}+z_{3}\right|$,
$\left|z_{2}\right|+\left|z_{4}\right|-\left|z_{2}+z_{4}\right|$,
$\left|z_{2}\right|+\left|z_{5}\right|-\left|z_{2}+z_{5}\right|$,
$\left|z_{3}\right|+\left|z_{4}\right|-\left|z_{3}+z_{4}\right|$,
$\left|z_{3}\right|+\left|z_{5}\right|-\left|z_{3}+z_{5}\right|$
and $\left|z_{4}\right|+\left|z_{5}\right|-\left|z_{4}+z_{5}\right|$is
greater than $8/25$?
Thanks!

I think the answer is no.
Suppose we can find such $z_1,\dotsc,z_5$. Then all of them are non-zero. Let $\theta_{ij}$ be the angle between $z_i$ and $z_j$. 
For all $i,j$, we have \begin{align*} \frac{8}{25} & \leq |z_i| + |z_j| - |z_i + z_j| \\ & = \frac{(|z_i| + |z_j|)^2 - |z_i + z_j|^2}{|z_i| + |z_j| + |z_i + z_j|} \\ & \leq \frac{2|z_i||z_j|(1-\cos{\theta_{ij}})}{2|z_j|} \\ & = |z_i| (1-\cos{\theta_{ij}}).\end{align*}
Reorder $z_1,\dotsc,z_5$ so that $|z_1|\leq \dotsb \leq |z_5|$. Now as $|z_1| \leq \frac{1}{5}$, the above gives $\theta_{1j} \geq \cos^{-1}\left(-\frac{3}{5}\right)=2.21$ rad. This means if we join each $z_i$ to the origin separately, there is a region of $4.42$ rad around $z_1$ without any other $z_i$'s. So $z_2,\dotsc,z_5$ must be within a region of at most $2\pi - 4.42 = 1.87$ rad. (I rounded up/down frequently so that the bounds actually become cruder at each stage.)
In particular, there is some $i \neq 1$ such that $\theta_{ij} \leq \frac{1.87}{3} = 0.63$ radians, so that $|z_i| \geq 1.67$. Contradiction.
