Convex Hull is Smallest Convex Set containing Set/Corollary

Theorem
Let $X$ be a vector space over $\R$. Let $K \subseteq X$ be non-empty.

Then:


 * $K$ is convex $\map {\operatorname {conv} } K = K$

where $\map {\operatorname {conv} } K$ denotes the convex hull of $K$.

Sufficient Condition
Suppose that:


 * $\map {\operatorname {conv} } K = K$

From Convex Hull is Smallest Convex Set containing Set, we have:


 * $\map {\operatorname {conv} } K$ is convex.

So:


 * $K$ is convex.

Necessary Condition
Suppose that:


 * $K$ is convex.

From Convex Hull is Smallest Convex Set containing Set, we have:


 * $K \subseteq \map {\operatorname {conv} } K$

Note that $K$ is a convex set with $K \subseteq K$, from Set is Subset of Itself.

So Convex Hull is Smallest Convex Set containing Set also gives:


 * $\map {\operatorname {conv} } K \subseteq K$

so:


 * $\map {\operatorname {conv} } K = K$