Supporting Hyperplane of a Subset of $\mathbb{R}^n$ at a Point

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

(i) Supporting hyperplane. $H(a,\alpha)$ supports $C$ at $x$ if $x\in C$, $x\in H(a,\alpha)$, and $C\subseteq H^{\le}(a,\alpha)$ or $C\subseteq H^{\ge}(a,\alpha)$.