I am reading the book "Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory" by Francesco Maggi. In the chapter 7, called Lipschitz functions. There, in order to prove Kirzbraun's theorem he presents lemma 7.4 which states the following.
Lemma 7.4.
Given a finite collection of closed balls $(B[x_k,r_k])_{k=1}^N$ in $\mathbb{R}^n$, set \begin{equation*} C_t = \bigcap_{k=1}^N B[x_k,tr_k], \text{ for } t \geq 0 \end{equation*} If $s = \inf \{t\geq 0: C_t \neq \emptyset \}$, then $s < \infty$ and $C_s$ reduces to a single point $x_0$, which belongs to the convex hull of those $x_k$ such that $|x_0-x_k|=sr_k$.
He divides the proof in two steps, the first step proves that $C_s = \{x_0\}$ and the second step proves thats $x_0$ belongs to the convex hull of the $x_k$ such that $|x_0 - x_k| = sr_k$.
I understand the first step so I won't type it.
Proof: step 2.
Up to permuting the $x_k$, we may assume that $|x_k-x_0|=sr_k$ if and only if $1 \leq k \leq M$ for some $M \leq N$. Replacing $\{B[x_k,r_k]\}_{k=1}^N$ with $\{B[x_k,r_k]\}_{k=1}^M$, we do not change $s$ and $x_0$. We thus assume that $|x_k-x_0| =sr_k$ for $k=1,\dots,N$.
If now $v \in S^{n-1}$ (i.e $|v| = 1$) and $\epsilon > 0$, then by construction of $x_0$ there exists $x_k$ with \begin{equation*} |x_k-x_0|^2 < |x_k-(x_0+\epsilon v)|^2 = |x_k-x_0|^2 + \epsilon^2-2\epsilon v \cdot (x_k-x_0) \end{equation*}
i.e $2v\cdot (x_k-x_0) \leq \epsilon$. Hence, for every $v \in S^{n-1}$ there exists $x_k$ such that $v\cdot(x_0-x_k) \geq 0$. As a consequence, for every closed half space $H$ with $x_0 \in \partial H$ there exists $k \in \{1,\dots,N\}$ such that $x_k \in H$, that is, $\{x_0\}$ and $\{x_k\}_{k=1}^N$ cannot be separated by a hyperplane. (End of proof)
What I don't understand is why " As a consequence, for every closed half space $H$ with $x_0 \in \partial H$ there exists $k \in \{1,\dots,N\}$ such that $x_k \in H$, that is, $\{x_0\}$ and $\{x_k\}_{k=1}^N$ cannot be separated by a hyperplane." implies that $x_0$ is in the convex hull of the $\{x_k\}_{k=1}^N$. If someone could clarify that to me, I would be very grateful.
Pd: In the book, for the closed balls, the notation $\overline{B}(x_k,r_k)$ is used. I changed to $B[x_k,r_k]$ because I prefer that latter. By the way, I understant that Kirsbraun's theorem hold for Hilbert spaces but I haven't been able to find a proof for hilbert spaces, does anyone know a reference?
Let $K$ be the convex hull of $\{x_k\}_{k=1}^m$. If $x_0 \notin K$ then the hyperplane separation theorem (also known as the finite-dimensional Hahn Banach Theorem) guarantees that there is a vector $z \in S^{n-1}$ such that $z \cdot x_0 < z \cdot y$ for all $y \in K$. Indeed, if $w \in K$ is the element of $K$ closest to $x_0$, then we can take $$z:=\frac{w-x_0}{\|w-x_0\|_2} \,.$$
You can find the proof in almost every book on convexity, and also in Theorem 2.6.2 (page 45) of [1]. For the case of polytopes (pertinent here) there is a proof via Farkas lemma in [2]. See also "Separation Theorem I" in [3].
[1] https://www.yuval-peres-books.com/game-theory-alive/ https://homes.cs.washington.edu/~karlin/GameTheoryBook.pdf
[2] https://terrytao.wordpress.com/2007/11/30/the-hahn-banach-theorem-mengers-theorem-and-hellys-theorem/
[3] https://en.wikipedia.org/wiki/Hyperplane_separation_theorem