See the Lemma 6.3 in RCA by Rudin:
Lemma 6.3 Let $z_1$, ... , $z_N$ be complex numbers. Then There is a subset $S$ of $\{1,\,\cdots ,\, z_N\}$ such that $$ \left| \sum_{k\in S} z_k \right| \ge \frac {1}{\pi} \sum_{k=1}^{N} |z_k|.$$
I have an example which shows that the constant $1/\pi$ is sharp.
Example. Let $N$ be an even positive integer. Let $\zeta := e^{2\pi i / N}$ and $z_k := \zeta^k - \zeta ^{k-1}$ for each $k=1, ... , N$. If $\epsilon$ is an arbitrary positive number, then we may say that $$ 2-\epsilon < \frac {1}{\pi} \sum_{k=1}^{N} |z_k| < 2 = \left| \sum_{k=1}^{N/2} z_k \right|$$ by growing $N$ large enough.
But the equality in Lemma 6.3 does not hold in this case.
Question: Is there any example which makes the equality hold?
