The Closed Convex Hull of a Set in a LCTVS

Definition: Let $X$ be a locally convex topological vector space and let $K$ be a subset of $X$. The Closed Convex Hull of $K$ is the smallest closed convex set containing $K$.

Equivalently, if $\{ E_i : i \in I \}$ is the collection of all closed convex sets contains $K$ then the closed convex hull of $K$ is $\displaystyle{\bigcap_{i \in I} E_i}$.

The closed convex hull of some sets in $\mathbb{R}^2$ are given below: