Supporting hyperplane theorem

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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 $α$ such that

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

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This article has been tagged since February 2007.