Convex hull via convex combinations

Theorem. Let $\Omega\subset X$ be a nonempty subset of a real vector space $X$. Then the convex hull can be described as

Context. This gives a constructive description of convex hulls: start from points in $\Omega$ and take all possible convex combinations .

Proof sketch. Let $C$ be the set of all finite convex combinations of points in $\Omega$. By closure under convex combinations , $C$ is convex and contains $\Omega$, hence $\mathrm{co}(\Omega)\subset C$ by minimality. Conversely, any convex set containing $\Omega$ must contain all convex combinations of points in $\Omega$, so $C\subset \mathrm{co}(\Omega)$.