# Proof of Theorem 18.8 in Rockafellar's Convex Analysis

Theorem: An $n$-dimensional closed convex set $C$ in $\mathbb{R}^n$ is the intersection of the closed half-spaces tangent to it.

In the beginning of the proof, it is established that the epigraph $G$ of the support function of $C$ is the closure of the convex hull of the set $S$ of all exposed rays of $G$, i.e., $G = \operatorname{cl}(\operatorname{conv}(S))$. But then Rockafellar says that "...it follows that...$C$ is the intersection of all the half-spaces $\left\{x | \langle x, x^* \rangle \le \alpha \right\}$, such that the set of non-negative multiples of $(x^*,\alpha)$ is a non-vertical exposed ray of $G$." How do we see that? Why are the vertical exposed rays (if any) not considered?

## 1 Answer

If it's ok, I'll postpone the discussion of vertical exposed rays until the end. First, about the non-vertical ones. Rockafellar states

'...the linear functions $\langle x, \cdot \rangle$ majored by $\delta^*(\cdot|C)$, which correspond of course to points in $C$, are the same as the linear functions whose epigraphs contain every non-vertical exposed ray of G.'

Let $NV\subseteq S$ be the 'non-vertical' exposed rays of $G$. For each ray $r$ in $NV$ is a non-vertical ray, we may write it as $r := r(x^*, \alpha):= \{\lambda (x^*, \alpha): \lambda\geq 0\}$ for some non-zero $x^*\in \mathbb{R}^n$ and $\alpha\in \mathbb{R}$. Note that $x^*\neq 0$ which comes from the ray being non-vertical.

With this notation, \begin{align*} C = &\{x\in \mathbb{R}^n: \langle x, x^*\rangle\leq \alpha,~ \forall ~r(x^*, \alpha)\in NV\}\\ =&\bigcap_{r(x^*, \alpha)\in NV}\{x\in \mathbb{R}^n: \langle x, x^*\rangle\leq \alpha\}. \end{align*}

Hence, $C$ is the intersection of half-spaces of the form $\{x\in \mathbb{R}^n: \langle x, x^*\rangle\leq \alpha\}$ where $r(x^*, \alpha)$ is a non-vertical extreme ray. In other words, $C$ is the intersection of half-spaces of the form $\{x\in \mathbb{R}^n: \langle x, x^*\rangle\leq \alpha\}$ where all non-negative scalars of $(x^*, \alpha)$ form a non-vertical extreme ray of $C$.

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Now for vertical exposed rays, I'll try to give an intuition (I hope this helps). A vertical exposed ray of $G$ is defined by a vertical hyperplane $H$ in $\mathbb{R}^{n+1}$ that defines a valid inequality for $G$ --- by a vertical hyperplane, I mean one of the form $H=\{(x, y)\in \mathbb{R}^n\times \mathbb{R}: \langle (x,y), (x^*, 0)\rangle = \beta\}$ for some $x^*\in \mathbb{R}^n$ and $\beta\in \mathbb{R}$ (note that $\beta$ will actually be $0$ here because of the form of a vertical ray). Note that $G$ is contained on one side of $H$. On the side of $H$ that does not contain $G$, the function support function $\delta^*(\cdot|C)$ equals infinity. This means that if $x\in \mathbb{R}^n$ is separated from $G$ by $H$, then $\delta^*(x|C) = \sup\{\langle x, x^*\rangle:~x^*\in C\} = \infty$ and so $x$ is a recession direction of $C$. So vertical exposed rays do not need to be considered as they define unbounded directions of $C$.

• Thanks a lot! Please allow for some time to go carefully through your answer. – Manos Feb 1 '17 at 4:01
• You can see how vertical rays are unnecessary from the previous equations for defining $C$. A vertical ray is of the form $r(0, \alpha)$. If you substitute this into the description of $C$, we obtain $$C = \left(\bigcap_{r\in NV}\{x\in \mathbb{R}^n: \langle x, x^*\rangle \leq \alpha\}\right)\cap \left(\bigcap_{r\not\in NV}\{x\in \mathbb{R}^n: \langle x, 0\rangle \leq \alpha\} \right)$$ The latter sets are just $\mathbb{R}^n$, so they don't `contribute' to $C$ – JSP Feb 1 '17 at 4:09