New Definition of Convex Hull I have always defined the convex hull of a set $X$ to be the smallest convex set that contains $X$. 
I am currently reading about holomorphic convexity, and the author has introduced a new definition of the convex hull. That is, the convex hull of a set $X$ is the intersection of all half spaces containing $X$, or, in other words, it is the set of points at which any real linear function takes values not exceeding its maximum on $X$. 
Can someone help characterise the equivalence of these three definitions? 
 A: This is in a locally convex topological vector space over $\mathbb R$.  The equivalence is for the closed convex hull, using closed half-spaces, closed convex sets,
and  continuous linear functionals. That is, the closed convex hull of $X$ is the intersection of all closed convex sets containing $X$, and this is also the intersection of all closed half-spaces containing $X$.
Counterexample to your statement: In $\mathbb R^2$, let $X$ be the union of the open half-space $\{(x,y): x > 0\}$ and the point $(0,0)$.  This is a convex set; its convex hull is itself.  But any half-space containing $X$ contains the whole $y$ axis, so the intersection of these half-spaces is $\{(x,y): x \ge 0\}$.
A closed half-space is a closed convex set, and the intersection of closed convex sets is a closed convex set, so the intersection of all half-spaces containing $X$ is a closed convex set containing $X$.
Conversely, if $C$ is a closed convex set and $p$ a point not in $C$, the 
Hahn-Banach separation theorem says there is a closed half-space containing $C$ but not $p$.  So any point in the intersection of all closed half-spaces containing $X$ is in the closed convex hull of $X$.
