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 if and only if $\map {\operatorname {conv} } K = K$

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

Proof

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.

$\Box$

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:

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

$\blacksquare$