Separating Hyperplanes in $\mathbb{R}^n$: Separation, Strict Separation and Strong Separation

In the setting of Separation in $\mathbb{R}^n$: Standing Notation, let $C,D\subseteq\mathbb{R}^{n}$, let $a\in\mathbb{R}^{n}$ with $a\neq0$ and let $\alpha\in\mathbb{R}$, with the hyperplane $H(a,\alpha)$, its closed half-spaces $H^{\le}(a,\alpha)$, $H^{\ge}(a,\alpha)$ and its open half-spaces $H^{<}(a,\alpha)$, $H^{>}(a,\alpha)$.

(i) Separation. $H(a,\alpha)$ separates $C$ and $D$ if $C\subseteq H^{\le}(a,\alpha)$ and $D\subseteq H^{\ge}(a,\alpha)$, or $C\subseteq H^{\ge}(a,\alpha)$ and $D\subseteq H^{\le}(a,\alpha)$.

(ii) Strict separation. $H(a,\alpha)$ strictly separates $C$ and $D$ if $C\subseteq H^{<}(a,\alpha)$ and $D\subseteq H^{>}(a,\alpha)$, or $C\subseteq H^{>}(a,\alpha)$ and $D\subseteq H^{<}(a,\alpha)$.

(iii) Strong separation. $H(a,\alpha)$ strongly separates $C$ and $D$ if there is a real number $\varepsilon>0$ such that

$$a\cdot x\le\alpha-\varepsilon\quad\text{for every }x\in C\qquad\text{and}\qquad a\cdot y\ge\alpha+\varepsilon\quad\text{for every }y\in D,$$

or the same holds with $C$ and $D$ exchanged.