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 
$$C\_d = \frac{\Gamma(\tfrac{d}{2})}{2 \sqrt{\pi}\, \Gamma(\tfrac{d+1}{2})}$$
where $\Gamma$ is Euler's Gamma function, so $C\_1 = \tfrac1{2}$, $C\_2 = \tfrac1{\pi}$ and these two are optimal.


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](http://mathworld.wolfram.com/SphericalCode.html).


Thank you.



Subset of vectors whose sum has a large norm
Finally, I found that a stronger result is known:


**Theorem.** For any integer $d \geq 1$ and $\delta \in [0,1]$, let
$$C\_{d,\delta} = \frac{\Gamma(\tfrac{d}{2})(1-\delta^2)^{(d-1)/2}}{2\sqrt{\pi}\,\Gamma(\tfrac{d+1}{2})}.$$
Then for all $v\_1, \ldots, v\_n \in \mathbb{R}^d$ there exists $u \in \mathbb{R}^d$ with $\|u\| = 1$ such that if $S$ is the set of $k \in \{1,\ldots,n\}$ with $v\_k \cdot u \geq \delta \|v\_k\|$, then
$$\left\|\sum\_{k \in S} v\_k\right\| \geq C\_{d,\delta} \sum\_{k=1}^n \|v\_k\|.$$
Moreover, the constant $C\_{d,\delta}$ is optimal.


I. Netuka, and J. Veselý - An inequality for finite sums in $\mathbb{R}^m$. *Časopis pro pěstování matematiky **103** (1978), p. 73--77.*


