Closed convex hull
= Closed convex hull
{title2=$\overline{\operatorname{co}}(C)$}
The smallest closed <convex set> containing a set $C$, equivalently the closure of its <convex hull> in the specified <topology>. The topology must be stated: the <norm topology>, <weak topology> and <weak-star topology> need not give the same closure for an arbitrary set.
Even a bounded infinite input need not have a closed <convex hull>. In the real line, let $S=\{1/n:n\geq1\}$. Every finite <convex combination> of these positive points is positive, so zero is absent. Conversely, any $x\in(0,1]$ lies between some $1/n$ and $1$, and is their <convex combination>. Hence
$$
\operatorname{conv}(S)=(0,1],\qquad\overline{\operatorname{conv}}(S)=[0,1].
$$
The <closed convex hull> adds the missing <limit> point zero. For the finite eight-point example in <convex hull>, the <convex hull> is already closed, so taking the <closed convex hull> adds nothing.