$\sum _1 ^n |z_j| \ge 1 \Rightarrow | \sum _1 ^k z_{j_m}| \ge C$

Question

Prove that there exists $C > 0$ such that the following implication holds:

If $\{z_1, ..., z_n \} \subset \mathbb{C}$ are such that $\sum _{j=1} ^n |z_j| \ge 1$,

then there exists $ \{z_{j_1}, ..., z_{j_k} \} \subset \{z_1, ..., z_n \}$ such that $ |\sum _{m=1} ^k z_{j_m}| \ge C$

What is the value of $C$?

Could you help me deal with this problem?

I'll be grateful for all your hints.

Answer 1

The answer, for the optimum value $C$ is $$ C=\frac{1}{\pi}. $$ This is Lemma 6.3, page 118, in W. Rudin's Real and Complex Analysis. This was first proved by Kaufmann and Rickert in Bull. Amer. Math. Soc., 72, p. 672-676, 1966.