Supporting hyperplane

In geometry, a supporting hyperplane of a set ${\displaystyle S}$ in Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ is a hyperplane that has both of the following two properties:

Here, a closed half-space is the half-space that includes the points within the hyperplane.

This theorem states that if ${\displaystyle S}$ is a convex set in the topological vector space ${\displaystyle X=\mathbb {R} ^{n},}$ and ${\displaystyle x_{0}}$ is a point on the boundary of ${\displaystyle S,}$ then there exists a supporting hyperplane containing ${\displaystyle x_{0}.}$ If ${\displaystyle x^{*}\in X^{*}\backslash \{0\}}$ (${\displaystyle X^{*}}$ is the dual space of ${\displaystyle X}$, ${\displaystyle x^{*}}$ is a nonzero linear functional) such that ${\displaystyle x^{*}\left(x_{0}\right)\geq x^{*}(x)}$ for all ${\displaystyle x\in S}$, then

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