Convex Hull of a Subset of $\mathbb{R}^n$

In the setting of Convex Sets in $\mathbb{R}^n$: Standing Notation, let $S\subseteq\mathbb{R}^{n}$.

(i) Convex hull. The convex hull $\operatorname{conv}S$ of $S$ is the unique convex subset of $\mathbb{R}^{n}$ that contains $S$ and is contained in every convex subset of $\mathbb{R}^{n}$ containing $S$; it exists and is unique by Existence and Uniqueness of the Smallest Convex Superset of a Subset of $\mathbb{R}^n$ (i).