Strong Separation of a Point from a Closed Convex Subset of $\mathbb{R}^n$

In the setting of Separation in $\mathbb{R}^n$: Standing Notation, let $C\subseteq\mathbb{R}^{n}$ be closed and convex, and let $x\in\mathbb{R}^{n}\setminus C$.

(i) Separating inequality. There is $a\in\mathbb{R}^{n}$ with $a\neq0$ such that

$$a\cdot y\le a\cdot x-\lVert a\rVert^{2}<a\cdot x\qquad\text{for every }y\in C.$$

(ii) Strong separation. For every $a$ as in (i), the hyperplane $H(a,\alpha)$ with $\alpha=a\cdot x-\tfrac{1}{2}\lVert a\rVert^{2}$ strongly separates $C$ and $\{x\}$.