Consider the following problem:
Prove that for every set of complex numbers $\{z_i\}$, with $i$ ranging from one to $n$, there is a subset $J$ such that
$$\left|\sum_{j\in J} z_j\right|\ge \frac{1}{4\sqrt 2} \sum_{k=1}^n |z_k|.$$
Could someone give me an example of equality? I believe I proven have a stronger statement.
My proof. Consider all the $z_i$ with positive real part. Call the real part of the sum of these numbers $X^+$. In a similar way, form $X^-$, $Y^+$, and $Y^-$. Without loss of generality, let $X^+$ have the greatest magnitude of these.
Note that because $|\operatorname{Re}(z)|+|\operatorname{Im}(z)|\ge |z|$, we have
$$ \left(\sum_{k=1}^n |\operatorname{Re}(z_k)|+|\operatorname{Im}(z_k)| \right) \ge \sum_{k=1}^n |z_k|.$$
But note that $\sum \limits_{k=1}^n |\operatorname{Re}(z_k)|+|\operatorname{Im}(z_k)| = X^+ + |X^-|+ Y^+ +|Y^-|$, so we have $$ 4X^+ \ge \sum_{k=1}^n |z_k|.$$ By choosing $J$ to be the set of complex number with positive real part, this proves a stronger statement, because the factor of $1/\sqrt 2$ isn't needed.