Exercise 1, § 3

a) Let $(x_k)_{1 \leq k \leq m}$ be a finite sequence of real numbers such that $|x_k| \leq 1$ for $1 \leq k \leq m$ and $\sum_{k=1}^m x_k = 0$. Show that there exists a permutation $\sigma$ of the interval $[1, m]$ of $\mathbf{N}$ such that

for all $p = 1, 2, \ldots, m$, and which preserves the order of the indices $h$ for which $x_h > 0$ and the order of the indices $h$ for which $x_h < 0$ [i.e., if $h < k$ and $x_h > 0$ and $x_k > 0$, then $\sigma(h) < \sigma(k)$, and similarly for indices $h$ such that $x_h < 0$].

b) Let $(x_k)_{1 \leq k \leq m}$ be a finite sequence of points $x_k = (x_{ki})_{1 \leq i \leq n}$ of $\mathbf{R}^n$, such that $||x_k|| \leq 1$ for $1 \leq k \leq m$, and $\sum_{k=1}^m x_k = 0$. Show that there exists a permutation $\sigma$ of the interval $[1, m]$ of $\mathbf{N}$ such that

[Proof by induction on $n$, considering $\mathbf{R}^n$ as a product $\mathbf{R}^{n-1} \times \mathbf{R}$, and putting $x_k = (x'_k, x_{kn})$ with $x'_k \in \mathbf{R}^{n-1}$. Take a subset $H$ of the interval $[1, m]$ of $\mathbf{N}$ such that $\left\|\sum_{k \in H} x_k\right\|$ is a maximum; by means of a rotation, reduce to the case where $\sum_{k \in H} x'_k = 0$ and show that in this case we must have $x_{kn} \geq 0$ for $k \in H$ and $x_{kn} \leq 0$ for $k \notin H$; then use a) and the inductive hypothesis.]

c) Let $(x_k)_{1 \leq k \leq m}$ be a finite sequence of points of $\mathbf{R}^n$ such that $\|x_k\| \leq 1$ for $1 \leq k \leq m$ and $\left\|\sum_{k=1}^m x_k\right\| = a > 0$. Show that there exists a permutation $\sigma$ of the interval $[1, m]$ of $\mathbf{N}$ such that

for all $p$ such that $1 \leq p \leq m$ [reduce to case b)].