In Rudin - Real & Complex Analysis we have the following

Lemma 6.3. If $z_1, \ldots, z_n \in \mathbb{C}$ then there is a subset $S \subseteq \{1,\ldots,n\}$ for which $$\left|\sum_{k \in S} z_k\right| \geq \frac1{\pi} \sum_{k=1}^n |z_k|.$$

This is used by Rudin to prove that a complex measure $\mu$ has a bounded total variation $|\mu|$. This lemma is presented also as an exercise in Krantz - Techniques of Problem Solving (Chap. 2, Ex. 34) and Greene and Krantz - Function Theory of One Complex Variable (Chap. 1, Ex. 57). Moreover, considering $z_k = e^{2\pi i k / n}$, $k=1,2,\ldots,n$, as $n \to \infty$, one can prove that the constant $\frac1{\pi}$ is in fact optimal.

Reading Rudin proof, it turn out that the algebraic structure of $\mathbb{C}$ plays no special role, so that $\mathbb{C}$ is considered as $\mathbb{R}^2$. Actually, without too much difficulty, one can adapt the proof to show the following

Theorem 1. For any integer $d \geq 1$ there exists an absolute constant $C_d > 0$, such that if $v_1, \ldots, v_n \in \mathbb{R}^d$ then there exists a subset $S \subseteq \{1, \ldots, n\}$ for which $$\left\|\sum_{k \in S} v_k\right\| \geq C_d \sum_{k=1}^n \|v_k\|,$$ where $\|\cdot\|$ is the euclidean norm of $\mathbb{R}^d$. In particular, we can take $C_1 = \tfrac1{2}$ and $C_d = \tfrac1{\pi}$ for $d \geq 2$.

In conclusion, my questions are: Are there some references for Theorem 1, and these kind on inequalities, in the literature? What can we say about the best constants $C_d$? And about the size of $S$?

I think the topic is related in some ways to spherical codes.

Thank you.