If $x = \sum_{i = 1}^n a_i y_i$ with $a_i > 0$, $\sum_{i = 1}^n a_i = 1$ and $|x| \geq |y_i|$ why is it true that $x = y_i$ for all $i = 1, \ldots, n$?
I can see that $|x| \leq \sum_{i = 1}^n a_i |y_i| \leq \sum_{i = 1}^n a_i |x| = |x|$ which means that $|x| = \sum_{i = 1}^n a_i |y_i|$. But how to I conclude?