Prove existence and uniqueness of Convex Hull containing compact set I want to prove the existence and uniqueness of the convex set described below, which is the convex hull.  My thinking is that I'm to generate a set containing all the convex sets containing $A$ and take their intersection.  Then pointing out that the intersection will also be convex.  How could I formalize the set containing all such convex sets containing $A$?  
Thanks in advance

If $A\subset\mathbb{R^n}$ is compact, then show that $\exists$ a unique convex subset $B$ of $\mathbb{R^n}$ such that $A\subset B$ and $B$ lies in any compact convex subset of $\mathbb{R^n}$ containing $A$.


To formalize: let $\mathscr C$ be the collection of sets (so it is a subset in the power set of $\mathbb R^n$): 
$$\mathscr C:= \{ B\subset \mathbb R^n : B \text{ is convex and } A\subset B\}.$$
Then the set you want is 
$$ A^h := \bigcap _{B\in \mathscr C} B.$$
