Definition 13.1.21 (Separating Hyperplanes for Convex Sets in $\R^{n}$).label Let $H\suf\R^{n}$ be subsets then $H$ is a hyperplane if $\exists a\in\R^{n}\del \curl{\vec{0}}, \al\in\R$ such that $H=\curl{x\in\R^n:\inner{x}{a}=\al}$. Let $A,B\suf\R^{n}$ be subsets then a hyperplane $H:\inner{x}{a}=\al$ (equivalently, the pair $(a,\al)$) weakly separates $A$ and $B$ if $\fall x\in A,y\in B$

The following are equivalent

1. (CS1)

   $\exists \e>0$ such that

   \begin{align*}\inner{x}{a}\leq \al-\e&&\inner{y}{a}\geq \al+\e\end{align*}

   
   
2. (CS2)

   $\SUP{x\in A}\inner{x}{a}<\INF{y\in B}\inner{y}{a}$

$H$ strongly separates $A$ and $B$ if it satisfies the above conditions.