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!