Supporting hyperplane theorem

The supporting hyperplane theorem is the mathematical theorem which states that every closed convex set in Euclidean space can be separated from an arbitrary point outside the set by a hyperplane.

More precisely, suppose that $C$ is a closed convex set in $\mathbb {R} ^{n}$ and that $y$ is a vector in $\mathbb {R} ^{n}$ that is not in $C$. Then there is a nonzero vector $a\in \mathbb {R} ^{n}$ and a real number $\alpha$ such that

$a\cdot x\leq \alpha \leq a\cdot y\quad {\text{for all }}x\in C.$